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Barrier Properties of Polymer Nanocomposites† Lakshmi N. Sridhar* Department of Chemical Engineering, UniVersity of Puerto Rico, Mayaguez, Puerto Rico 00681-9046
Rakesh K. Gupta and Mohit Bhardwaj Department of Chemical Engineering, West Virginia UniVersity, P.O. Box 6102, Morgantown, West Virginia 26506
A computationally efficient strategy, based on a resistance-in-series and resistance-in-parallel approach, is developed for obtaining diffusivities when nanometer-sized flakes are introduced in a polymer matrix. The mass-transfer resistance is properly defined, and several examples are presented to demonstrate the reliability of this technique. One is also able to determine the optimum configuration to minimize the diffusivity by maximizing the resistance in the polymer matrix. The extent of diffusivity reduction with variables such as the filler loading level and filler geometry is explored and explained. Comparison is also made with a twodimensional model where the Laplace differential equation is solved using a finite difference method, and agreement with experimental results demonstrates the effectiveness of this technique. Introduction Polymer nanocomposites (PNCs) formed by dispersing a few weight percent of nanometer-sized fillers, such as carbon nanotubes and montmorillonite (clay), in either thermoplastic or thermosetting polymers are now commercially available. Compared to neat polymers, PNCs have a tendency to have higher tensile and flexural moduli, improved barrier properties, and enhanced flame resistance. In terms of barrier properties, a reduction in the diffusion coefficient is thought to result simply from the increase in path length that is encountered by a diffusing molecule, because of the presence of an enormous number of (passive) barrier particles during mass transfer. This principle has been utilized to develop better tennis balls, improved packaging for juice and beer, and protective coatings for fuel and chemical tanks.1 Here, we examine diffusion through PNCs in detail, and we seek to quantitatively relate the reduction in diffusivity to the morphology of the nanocomposite. Diffusion through a heterogeneous, two-phase medium can be expected to be dependent on the properties of the individual phases, and the amount, shape, size, size distribution, and orientation of the dispersed phase, and early work on diffusion through arrays of spheres, cylinders, and ellipsoids has been summarized by Crank.2 Barrer3 studied diffusion through two types of heterogeneous media: (i) laminates, where layers of † This paper is submitted to the special issue of Industrial and Engineering Chemistry Research honoring Dr. Warren Seider on the occasion of his 65th birthday. This particular section is Dr. Sridhar’s tribute to Dr. Seider. Dr. Sridhar wishes to state that “...Dr. Seider has been a role model and a source of tremendous inspiration to me, not only as an awesome academician but also as a wonderful human being. In an atmosphere where grant chasing seems of primary importance, Dr. Seider has demonstrated the importance of teaching and imparting knowledge to both undergraduate and graduate students. This attitude is reflected in his textbook Process Design Principles, which enables undergraduate students to learn the principles of process design without too much aid from an instructor. Despite his academic achievements and a hectic schedule, Dr. Seider has always taken time to advise younger faculty like me on various academic and nonacademic issues. I therefore feel tremendously honored to be able to submit an article for this special issue.” * To whom all correspondence should be addressed. Tel: (787) 8324040. Fax: (787) 265-3818. E-mail:
[email protected].
different properties are sandwiched together, and (ii) particulate composites, where discrete particles of one phase are dispersed in a continuum of another. Although the mass-transfer resistance has often not been explicitly defined in the literature, the seriesparallel formulation to obtain the effective diffusivity has been proposed for both situations (i) and (ii) by several authors. In addition, Fidelle and Kirk4 and Bell and Crank,5 among other researchers, numerically solved the two-dimensional Laplace equation over a domain appropriate to case (ii) and compared the results with the expressions predicted by the series-parallel and parallel-series formulations. They found that the numerical results were bounded by the two analytical expressions. However, this approach does not seem to have been attempted for the situation where the filler is of nanometer size. Quantitatively predicting the reduction in diffusivity of a diffusing molecule in a polymer containing an impenetrable filler is an important problem that is of both fundamental and practical importance. On the fundamental side, Nielsen6 considered twodimensional diffusion through a polymer that contained infinitely long, rectangular-cross-section plates that were uniformly dispersed in the polymer but were placed normal to the direction of mass transfer. By calculating the maximum possible tortuosity factor, he determined that the largest possible ratio of the diffusivity of a molecule through the neat polymer (D0) to that of the same molecule through the filled polymer (D) was given as
D0 Rφ )1+ D 2
(1)
where φ is the volume fraction of filler and R is the aspect ratio of the rectangle that forms the barrier cross section (the latter parameter is defined as R ) w/(2t); see Figure 1). More recently, Cussler and co-workers7-12 have studied the diffusion problem through flake-filled membranes and developed an extensive theory for predicting the changes in diffusivity, again as a function of the loading level and aspect ratio of the filler particles. Their basic equation is
D0 R2φ2 )1+ D 1-φ
(2)
and the predictions of this equation can differ significantly from
10.1021/ie0510223 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/18/2006
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Figure 1. Morphology of a model polymer nanocomposite: (a) three-dimensional view of the rectangular cross-section of flakes distributed in a polymer matrix; (b) two-dimensional view of flakelike barriers in the x-z plane; and (c) same as panel b, but showing nanochips staggered by having alternate layers displaced in the x-direction. panel d shows a plot of the quantity θ, defined as s/w, which is used to measure the extent of stagger in the x-direction.
those of eq 1, even at low loading levels.13 This is especially true at large values of R, where the second term on the righthand side of the two equations becomes larger than unity. Thus, the expression for D0/D should involve more quantities than just R and φ, at least under some conditions. Note that in the derivation of both eqs 1 and 2, it is assumed that the barriers, which are in the form of flakes, have one dimension that is the same as the membrane width. Therefore, it is important to be able to determine how the diffusivity reduces when we introduce barriers in the form of “nanochips” or nanofillers whose ratio of length to thickness is comparable to their ratio of width to thickness. From a practical standpoint, polymer nanocomposites based on thermoplastic polymers are mass-produced by the process of extrusion compounding, wherein agglomerated clay platelets are sought to be dispersed in the matrix of a polymer such as polypropylene or nylon 6. Here, one wishes to know how the barrier properties of a nanocomposite are dependent on the filler loading level when the filler thickness, filler aspect ratio, filler orientation, and extent of filler dispersion are kept unchanged. One then wishes to know how the diffusivity changes when the loading level is kept constant but the aspect ratio is changed, either by changing the filler thickness or the filler length; the former situation arises when one goes from intercalated platelets to exfoliated platelets, whereas the latter situation arises when there is attrition of filler in the extruder. One also wants to know the effect of a distribution of filler sizes. We have attempted to answer some of the above questions in this paper. In support of the computational work, we present experimental data from other researchers, as well as our own data, on the diffusion of moisture through PNCs made with montmorillonite and vinyl ester. We have also attempted to determine the optimum configuration of embedded nanofillers required to minimize the diffusivity. Problem Statement The physical situation being considered is shown in Figure 1a, where mass transfer happens in the positive z-direction
through a membrane containing a very large number of rectangular-cross-section flakes or nanochips. Only three layers of flakes are shown, although the total number of layers is very large. As seen in the front view (the x-z plane) in Figure 1b, each flake has a thickness t and width w; the third dimension is also w. Thus, in the x-y plane, each flake is a square of area w2. The diffusion coefficient is reduced by the presence of the flakes; however, the reduction is likely to be the smallest when the nanochips are arranged below each other, as in Figure 1b, where each nanochip completely overlaps the chip below it. The diffusion coefficient reduces further when the chips are staggered in the x-direction, as shown in Figure 1c; the extent of overlap in the x-direction is measured via the quantity θ, which is defined as θ ) s/w (see Figure 1d). Clearly, D decreases as θ decreases from its initial value of unity; however, there is likely to be little additional decrease in D as the extent of stagger is increased beyond a value of θ ) zero, at least under some conditions. Of course, the diffusion coefficient can be reduced further by staggering the chips in the y-direction; however, this is not considered here. To completely define the morphology, we also need to specify three other quantities. With reference to Figure 1b, the intermediate length or the distance between neighboring flakes in the x-direction is taken to be l, and the vertical distance between two layers is the quantity T-t, whereas the distance between neighboring arrays of chips in the y-direction is h. Thus, a single chip is contained in a unit cell that is a rectangular parallelepiped of dimensions (l + w) in the x-direction, (w + h) in the y-direction, and T in the z-direction. As a consequence, the volume fraction of filler φ is (w2t)/[(l + w)(w + h)T]. Note that, for small values of l, it is possible for a chip to cross the boundary of a neighboring cell as θ is decreased or as the stagger is increased. From an analytical standpoint, therefore, the main objectives of this work are (i) to obtain the value of the nanocomposite diffusivity D for specified values of R, φ, θ, l, t, and T, and (ii) to obain the value of θ that minimizes D for given values of the other five variables.
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Figure 2. Identification of the various blocks used to compute the masstransfer resistance within the polymer nanocomposite.
Solution Procedure Since we have repetitions after two layers, we divide the region of interest into cells that repeat themselves in three dimensions. These cells lie in the portion between the dotted lines shown in Figure 2. Two different cells can be distinguished: the upper cell is designated “top intermediate block”, and the lower cell is designated “bottom intermediate block”. In addition, there is a “top end block” at both the left and the right, and a “bottom end block” at both the left and the right. Note that the bottom layer right end block has only one chip. Also note that the top layers in Figure 2 have undisplaced nanochips, whereas the chips in the bottom layer are displaced to the right. The resistance to mass transfer, which is defined as the ratio of the concentration driving force to the mass-transfer rate, for any of the cells described previously, can be calculated using the series parallel method. With reference to Figure 3, the resistance of block P (Rp) will be
RP )
Lp DpolAP
(3)
in which Dpol is the diffusion coefficient for the neat polymer. Similarly, the resistance of the chip will be
Rchip )
t DclayAP
(4)
where Dclay is the diffusion coefficient for clay, and its value is assumed to be very close to zero. Again, the resistance of block Q will be
LQ RQ ) DpolAP
(5)
and the resistance of the left block (RPQ) will be (RP + RQ + Rchip), because blocks P and Q and the chip are in series. The resistance of block R (RR) can be calculated in a similar manner, and the effective resistance of the entire block will be (1/RPQ + 1/RR)-1. Using this strategy, the resistances of the different cells shown in Figure 2 can be calculated. The entire top-layer resistance can be determined by treating the resistances of the left, right, and intermediate blocks of the top layer as resistances
Figure 3. Schematic of a representative block with an imbedded polymer chip.
Figure 4. Schematic path of least resistance of diffusing molecules when alternate layers of flakes are staggered with respect to each other.
in parallel and using the reciprocal resistance relationship
Rtop layer )
(
N 1 + + Rtop layer left block Rtop layer intermediate block 1 Rtop layer right block
)
-1
(6)
where N is taken to be a large number. Similarly, the resistance of the bottom layer also can be calculated. The total resistance of both the blocks will be
Rtotal ) Rtop layer + Rbottom layer
(7)
and the effective diffusivity D can be calculated as
D)
thickness of 2 layers (total area)(Rtotal)
(8)
from which the ratio of the two diffusivities (D0/D) can be easily calculated. This procedure is used when there is complete overlap. When there is only partial overlap between the chips in the top layer and the bottom layer, the mass-transfer path is that shown in Figure 4. The path length now is larger than before, while the area for diffusion is smaller than before; this is the reason the diffusivity decreases with increasing stagger. As the amount of stagger is increased, at small values of l, the diffusivity will go through a minimum, because further increases in stagger will bring one back to the starting configuration. At very large values of l, on the other hand, the diffusivity will decrease initially but then remain unchanged.
Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006 8285 Table 1. Values of the Diffusivity Ratio When r ) 10'
Table 2. Values of the Diffusivity Ratio When r ) 30'
Diffusivity Ratio, D0/D
Diffusivity Ratio, D0/D
φ
θ
σ ) 0.1
σ ) 1.0
σ ) 10
φ
θ
σ ) 0.1
σ ) 1.0
σ ) 10
0.05 0.05 0.05 0.05 0.05
0.0001 0.25 0.5 0.75 0.9999
201.00 214.201 251.75 213.68 201
21.05 23.31 26.79 222.45 21.00
4.68 6.79 4.68 3.42 3.00
0.05 0.05 0.05 0.05 0.05
0.0001 0.25 0.5 0.75 0.9999
601.0 944.72 1957.8 940.19 601.00
61.16 101.35 202.86 96.48 61.0
9.38 21.88 28.43 12.36 7.00
0.2 0.2 0.2 0.2 0.2
0.0001 0.25 0.5 0.75 0.9999
21.93 54.34 113.61 44.75 21.00
30.00 63.75 30.0 9.75 3.0
0.2 0.2 0.2 0.2 0.2
0.0001 0.25 0.5 0.75 0.9999
601.00 6100 22309 6028 601.0
63.55 706.6 2330.8 628.45 61.00
45.16 245.2 350.0 92.75 7.00
201.0 412.21 1013.1 404.01 201.00
We believe that this very simple approach of ours captures the essential physics of the problem, and the computational results allow us to reach useful conclusions. Note that although h is likely to be of the same order of magnitude as l, we have assumed its value to be zero in deriving eqs 6-8. This was done to compare our results with those of Cussler and coworkers. Also, by assuming h to be nonzero, the only additional effect that we could examine is the influence of staggering the flakes in the y-direction. The resulting conclusions would be identical to those obtained by staggering the flakes in the x-direction alone. Results The computational results are presented in dimensionless form in Tables 1-4. Because of the fact that we are interested in polymer nanocomposites, it is helpful to think in terms of a flake thickness of 1 nm. If the width w is taken to be 20 nm, the aspect ratio R becomes 10. If we assume the distance l to be 0.1 nm, the flakes are very close to each other, and the dimensionless intermediate length σ, which is defined as the ratio of the intermediate length l to the flake thickness, has a value of 0.1. At a low loading level of φ ) 0.05, the corresponding value of T then is 19.9 nm. The effect of keeping the upper layer of flakes usdisturbed (fixed R and σ) while staggering the lower layer (by changing θ at fixed T) or changing the layer spacing (by changing φ at fixed l) is explored in Table 1. The gap between neighboring flakes is very small; therefore, the resistance to mass transfer is large, and even when the flakes are situated below each other (θ ) 0.9999), the ratio D0/D is 201. When θ is decreased even slightly, the gap between the flakes is completely covered by the layer of flakes above and below a given layer. This results in a progressive increase in the path length and a progressive decrease in the mass-transfer area for the diffusing molecules with a concomitant increase in D0/D. As a consequence, D0/D increases until a given flake is directly below the gap in the upper layer of flakes; when θ ) 0.5, D0/D has a value of 251.75, which is close to the maximum value for this quantity. Further reductions in θ then lead to decreases in D0/D. When the loading level φ is increased, keeping the other quantities unchanged, the layer spacing (T t) decreases. Thus, at a value of φ ) 0.1, T ) 9.95 nm, whereas when φ ) 0.2, T ) 4.975 nm. Consequently, at a fixed value of θ, increasing φ makes the diffusion path more and more tortuous, except when θ is unity. As a result, D0/D is independent of φ (at fixed σ) when θ ) 1.0, but it increases and then decreases in an essentially symmetrical manner as θ is reduced; however, the maximum value of D0/D is dependent sensitively on the loading level φ. In other words, the parameters R and φ are, by themselves, not enough to determine D0/D theoretically, as suggested by eqs 1 and 2.
Table 1 also examines the effect of increasing the intermediate length l, using a σ value of 1.0. Now, the gap between the flakes is 10 times larger than the value used in Table 1. This results in a sharp and essentially proportionate reduction in D0/D; when θ is close to unity, D0/D ) 21. Changes in θ and φ lead to variations in D0/D that are qualitatively and proportionately similar to those observed earlier. When the horizontal distance l between neighboring flakes is taken to be half the flake width, D0/D further reduces to a value of 3 when all the flakes are arranged below each other. As θ approaches zero, the flakes in a given layer are again able to cover the gaps between the flakes in the layers above and below. Consequencely, D0/D increases and then decreases with decreasing θ. If l were to be increased by a further order of magnitude, no amount of stagger would completely cover the gap, and the very large variation of D0/D with θ would no longer be observed. In this case, the D0/D value at θ ) 1 would still give the smallest value of D0/D, and it would become maximum at θ ) 0, but remain unchanged thereafter. Furthermore, when θ has reached a value of zero, changing the aspect ratio (or having a distribution of flake widths) at constant flake thickness will not affect the results. Clearly, the computations of Table 1 suggest that experimental results for D0/D, at constant values of R and φ, can lie in a fairly wide range. Practically speaking, this reflects the influence of the extent of dispersion, and it happens because of reasonable variations in the values of l and T, corresponding to the same values of R and φ. This fact does not seem to haVe been recognized in the literature, and it is a key message of the present work. Table 2 examines the effect of changing the aspect ratio R from 10 to 30. If we assume that R decreases from 30 to 10 because of a decrease in the chip width w, due to attrition, then a comparison of Tables 1 and 2 shows that, at the same values of σ and φ, D0/D is very significantly smaller. This happens because the fractional area available for mass transfer is increased when w is decreased without decreasing l. On the other hand, we may assume that R is reduced from 30 to 10 because three chips with dimensions of w ) 60 nm and t ) 1 nm have become stuck together. In this case, if l is considered to be unchanged at 1 nm, say, the numbers in Table 2 should be compared with those in Table 1 at a value of σ that is onethird of that in Table 2. One now finds that decreasing the aspect ratio has a tendency to reduce D0/D, because of the aspect ratio effect just considered, but the reduction is less than expected, because it is partly offset by a decrease in the dimensionless gap between neighboring chips. Thus, at a fixed value of φ, changes in the aspect ratio R, which are caused by a decrease in w, are not equivalent to those resulting from an increase in t. Table 2 also looks at the effect of changing the extent of stagger at different fixed values of nanofiller loading. The trends
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Table 3. Values of the Diffusivity Ratio When r ) 50 and σ ) 0.1' φ
θ
D0/D
0.05 0.05 0.05 0.05 0.05
0.0001 0.25 0.5 0.75 0.9999
1001 2580 7269.8 2568.2 1001
0.2 0.2 0.2 0.2 0.2
0.0001 0.25 0.5 0.75 0.9999
1001 2627 10130 26706 1001
Table 4. Variation of the Optimum Value of θ with σ σ
θoptimum
σ
θoptimum
σ
θoptimum
0.1 0.2 0.5 0.8 1.0 2.0
0.4975 0.495 0.4875 0.48 0.475 0.45
3.0 4.0 5.0 6.0 7.0 8.0
0.425 0.4 0.375 0.35 0.325 0.3
9.0 9.5 10.0 10.5 11.0 12.0
0.275 0.2625 0.25 0.2375 0.225 0.2
Figure 6. Computations showing the decrease in diffusivity as a function of nanochip volume fraction (φ) and layer spacing (T - t).
are similar to those observed earlier in Table 1, but the values of D0/D are now much larger, demonstrating the strong effect of R in the presence of an overlap of one layer of chips by another layer of chips. This is further reinforced by the calculations presented in Table 3, where the R has been increased further to 50. Clearly, if the goal is to reduce diffusivity by a few orders of magnitude, this can be accomplished at fairly low loading levels simply by using flakes with a large aspect ratio, by reducing the gap between neighboring flakes in any given layer, and by staggering the flakes to cover the (small) gaps in the layer above and the layer below. Optimization The question now to be addressed is this: for given values of R, φ, and t, what is the optimal value of θ that can be obtained for various values of σ to maximize the value of D0/D? We pose this as a minimization problem of the objective D/D0, because there is an upper bound and a lower bound on θ, namely (0,1). We have used the MATLAB routine fmincon to determine the answer. This is a medium-scale optimization problem, and for these types of problems, fmincon uses a Sequential Quadratic Programming (SQP) method. In this method, a Quadratic
Figure 7. Time-lag curves for diffusion of moisture through a neat vinyl ester film at 37.8 °C and three values of relative humidity difference.
Programming (QP) subproblem is solved at each iteration. An estimate of the Hessian of the Lagrangian is updated at each iteration, using the BFGS formula. The optimum values of θ for various values of σ are presented in Table 4 where the aspect ratio used is 10. These agree with our physical intuition that D0/D should be a maximum when any given flake completely covers the gap in the layer above and the layer below. Finite-Difference Calculations
Figure 5. Schematic of the geometry used for finite-difference calculations of the steady-state concentration during two-dimensional diffusion through a model polymer nanocomposite.
Having examined the situation where the flakes are very close to each other, we now turn to the situation where the flakes are far apart. This is the more likely scenario for polymer nanocomposites that contain a few weight percent of nanofiller. For this purpose, a finite difference scheme was used to compute the steady-state concentration profiles by solving Laplace’s equation over the region shown in Figure 5; this unit cell repeats itself in two dimensions. The purpose of solVing the Laplace’s equation is to compare the finite difference calculations and the experimental data with our model, and this comparison
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Figure 8. Time-lag curves for diffusion of moisture through a clay-vinyl ester nanocomposite film containing 1 wt % clay and measured at 37.8 °C and three values of relative humidity difference.
is shown later in Figure 10. Continuity of concentration and flux was assumed at the interface between the two phases. As mentioned previously, the barrier cross section is a rectangle of dimensions w and t, resulting in an aspect ratio of R ) w/(2t). The volume fraction of filler is clearly given as φ ) wt/(WT). Computational results are presented in Figure 6, and it is seen that, for fixed values of w, t, R, and φ, the reduction in diffusivity also is dependent on the spacing between two adjacent barrier layers. In other words, as one adjusts W and T to keep the product WT constant, D0/D changes significantly. As explained earlier, this is understandable. For small values of T, W is so large that there is one-dimensional mass transfer and D ) D0(1 - 2w/W). At the other extreme, for large values of T, W is so small that it approaches the minimum value of 2w and mass
transfer effectively ceases. The significance of this result is that it is difficult to predict the observed diffusivity reduction theoretically, especially at low filler contents, because the layer spacing is not known a priori. Indeed, at nanofiller volume fractions of
