Polyamide Interface. 3

Contributions to the free energy include (a) the elastic energy due to .... The segment density inside each cell is calculated by superposition of seg...
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Macromolecules 2002, 35, 508-521

Entanglement Network of the Polypropylene/Polyamide Interface. 3. Deformation to Fracture Andreas F. Terzis,† Doros N. Theodorou,*,‡,§ and Alexander Stroeks| Department of Physics, University of Patras, GR 26500 Patras, Greece; Department of Chemical Engineering, University of Patras, GR 26500 Patras, Greece; Institute of Chemical Engineering and High-Temperature Chemical Processes, GR 26500 Patras, Greece; and DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands Received April 23, 2001; Revised Manuscript Received September 5, 2001

ABSTRACT: Recently (Macromolecules 2000, 33, 1385, 1396), we have proposed a novel algorithm for generating entanglement network specimens of interfacial polymeric systems. The specimens are created by sampling the configurational distribution functions derived from a self-consistent mean field lattice model. Although overstretched strands can be relaxed by a Monte Carlo (MC) method, the specimens generated are not in detailed mechanical equilibrium. In this paper, we develop a method for relaxing the network with respect to its density distribution, and thereby imposing the condition of mechanical equilibrium, without changing the network topology. Our method rests on minimizing a free energy function characterizing the network, with respect to the coordinates of all entanglement points and chain ends. Contributions to the free energy include (a) the elastic energy due to stretching of the chain strands and (b) the free energy due to the repulsive and attractive (cohesive) interactions between segments. To calculate the interaction energy, a simple cubic grid is superimposed on the network. The repulsive interactions are calculated within each grid cell. The attractive interactions are calculated from contributions between cells and within each cell. We first apply the free energy minimization procedure to a simple system (bulk polypropylene) and obtain results which agree satisfactorily with experimental data and serve as a basis for parametrizing the interaction model. The free energy minimization is then applied to mechanically relax computer specimens of a polypropylene/polyamide6 (PP/PA6) interfacial system, strengthened by PP chains grafted onto the PA6 surface. The fully relaxed networks serve as a starting point for the mesoscopic simulation of fracture phenomena, caused by the application of tensile stress perpendicular to the interface. The network is deformed at a constant strain rate and the network topology evolves according to elementary mechanical processes of chain scission, chain slippage, disentanglement, and reentanglement. Chain slippage across an entanglement point occurs according to a Zhurkov activated rate equation with parameters derived from viscosity data. Each cycle of the kinetic MC algorithm used to track the deformation process consists of the imposition of a small incremental strain on the network, relaxation to mechanical equilibrium, introduction of the micromechanical processes mentioned above, and again relaxation to mechanical equilibrium. The MC cycles are repeated until fracture occurs. Results of the fracture experiments confirm the assertion, previously founded on structural grounds, that optimal adhesion in the PP/PA6 system examined is achieved for a relatively low surface grafting density of 0.1 chains/nm2.

1. Introduction This work is the continuation of an effort1,2 to develop theoretical and computer simulation tools for predicting the structure and mechanical properties of immiscible polymer/polymer interfaces compatibilized by adding, or forming in situ with appropriate chemical reactions, a diblock copolymer in which each block is miscible with one of the two homopolymers.3,4 We focus on the PP/ PA6 system, which has been the subject of several experimental studies,4-6 compatibilized with the reaction product between PP-g-MA (maleic anhydridefunctionalized PP) and PA6. In our previous work,2 we have developed an algorithm in order to generate coarse-grained entanglement networks, representative of this interfacial system. This * To whom correspondence should be addressed at the University of Patras. Telephone: +3061-997398. Fax: +3061-993255. E-mail: [email protected]. † Department of Physics, University of Patras. ‡ Department of Chemical Engineering, University of Patras. § Institute of Chemical Engineering and High-Temperature Chemical Processes. | DSM Research.

coarse-grained representation is based on the idea that the ultimate mechanical properties of polymeric systems are directly related to entanglements among chains. To generate networks representative of the real PP/ PA6 interfacial system, we have respected the information provided by experimental data.5,6 We have considered a crystalline PA6 matrix (i.e., an equivalent “solid” substrate) on which the PP chains of the compatibilizer are terminally grafted, extending into a bulk PP phase (i.e., a collection of free PP chains). Adopting a coarse-grained (mesoscopic) model is dictated by limitations in computer time for tracking the length scales and time scales relevant to fracture phenomena. Actually, two levels of coarse-graining are utilized. First, a fully occupied lattice is used to represent the system of interest, the structure of the lattice model being determined by a self-consistent field (SCF) theory. Next, the structural information derived from the SCF theory is used to build an entanglement network model in continuous three-dimensional space. In the lattice model, each chain species is viewed as a sequence of Flory segments, each segment occupying exactly one site. The Flory segment size is determined

10.1021/ma010691z CCC: $22.00 © 2002 American Chemical Society Published on Web 12/13/2001

Macromolecules, Vol. 35, No. 2, 2002

from the mass density and contour length of the homopolymer species present, as detailed in ref 1. Chain stiffness is also faithfully represented by introducing appropriate intermolecular energetics. In ref 2, using the one-dimensional information provided by the SCF theory as a starting point, we developed an algorithm in order to generate large, three-dimensional networks representative of the interfacial region. In the network, a chain is represented as a sequence of two ends and a set of intermediate nodal points, the latter participating in entanglements. Two chains come together at an entanglement point. Chain ends and entanglement points are referred to collectively as nodes of the network. Our construction results in a set of nodes for each chain, where each node has a specific contour position along the chain, positional vector in threedimensional space, and pairing with another node if it participates in an entanglement. The piece of chain between two nodes is referred to as a strand. The generated networks have been relaxed with regard to stretching of the strands (see Monte Carlo methodology developed in ref 2), but not with regard to the density distribution; thus, they are likely to incorporate large unbalanced stresses associated with excluded volume and cohesive interactions. To relax the large density fluctuations and bring the system to mechanical equilibrium (i.e., a state in which the total force at every node is zero), we have to minimize (locally) an appropriately formulated Helmholtz free energy with respect to all nodal positions. In this work, we develop a methodology to estimate the free energy of the coarse-grained network models. Conformational and nonbonded interaction contributions are taken into account. The conformational part comes from the stretched strands. The nonbonded interaction contribution is divided into an attractive part (van der Waals forces) and a repulsive part (excluded volume interactions). To estimate these contributions, the local segment density is needed. Once the free energy function is known, mechanical equilibrium is imposed by minimization of the free energy with respect to the positions of all nodes. Deformation experiments are performed on the network at a constant strain rate by a kinetic Monte Carlo (KMC) scheme, inspired by the one developed in ref 7. In the course of a simulation of deformation to fracture, the topology of the entanglement network changes through the introduction of elementary events (e.g., slippage of a chain past an entanglement point, chain rupture) governed by rate expressions and rules which are based on the reptation picture of polymer dynamics. KMC simulations of simplified two-dimensional network models have proved quite successful for addressing the terminal properties of semicrystalline polymers in the bulk,7 thus providing justification for the use of representations cast in terms of localized entanglement points for modeling the large-scale deformation and fracture of dense systems of interpenetrating chains. Here, a much more detailed three-dimensional scheme is developed, in which excluded volume interactions are accounted for explicitly, mechanical equilibrium is imposed throughout the deformation process, and no a priori imposition of the Poisson ratio is required. Typical sizes of the network system generated and deformed in this work are 0.1 µm on each side. The number of chains can be as high as 20 000 and the

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Figure 1. Simple cubic grid superimposed on the network specimen.

number of entanglement points as high as 130 000. The simulation of larger systems is well possible with currently available parallel computational resources. 2. Network Free Energy General Assumptions. Our network consists of chains, represented by specific points (nodes, i.e., entanglements or ends) along their contour. All nodal points have specific spatial positions. The initial contour length between consecutive entanglement points on the same chain is specified by the entanglement molecular weight Me (in this work, as in ref 2, Me ) 6 kg/mol). Note that, in the PP/PA6 interfacial system, the value for Me could be different, as all grafted and some free chains are more ordered along the direction perpendicular to the interface. However, as a first approximation, we use the bulk value because of lack of an alternative approach. To relax the network without changing its topology and thereby impose the condition of mechanical equilibrium at the current topology, we need to minimize the network free energy with respect to the coordinates of the entanglement points and chain ends. Contributions to the free energy include (a) the entropic free energy due to stretching of the chain strands and (b) the free energy due to the repulsive (excluded volume) and attractive (cohesive) interactions between segments. To calculate the interaction energy, a simple cubic grid is superimposed on the network (see Figure 1). The grid, which remains unchanged throughout all simulations described here, partitions the network into small cubic regions or cells. The edge length, lg, of each cell is commensurate with the distance between entanglements in three-dimensional space. The repulsive interactions are calculated within each grid cell. The attractive interactions are calculated from contributions between cells and within each cell. Initially, to test our method, we have applied the free energy minimization procedure to a network representative of bulk isotactic PP at room temperature. With the parameters we chose for the representation of nonbonded interactions (see below), the results were found to agree satisfactorily with experimental values of the mass density, cohesive energy density, and bulk modulus. Then, the free energy minimization procedure was applied to mechanically relax the more complex PP/ PA6 interfacial system. Network Density Distribution. The local density is calculated in each cell of the simple cubic grid of spacing lg that is superimposed on the network. Each strand of the network contributes to the local density in a way which depends on the number of Flory (or Kuhn) segments in the strand and on the spatial positions of the ends of the strand.

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Figure 3. Local coordinate system around a strand. The grid shown is the one over which we perform the pretabulation of the local density.

Figure 2. Strand “segment density clouds”. The segment density inside each cell is calculated by superposition of segment clouds.

Thus, the number density of monomers in a cell indexed by W is given by a sum of contributions to that cell from each strand of the network (see Figures 2, 3)

FW )

(

) )

R Bi + R Bj B i|,nKij × ,|R Bj - R 2 i< j lg lg Θ - |Rx - RxW| Θ - |Ry - RyW| × 2 2 lg Θ - |Rz - RzW| 2

∫dRB ∑∑cijF RB -

[(

) (

(

)]

(1)

where the sums are over all nodes (entanglement points and ends) i and j and R B W is the position vector of the center of the cell. The length of the strand between nodes i and j is specified by nKij, the number of Kuhn segments. The connectivity factor cij is defined as

cij )

{

nmK if nodes i and j are connected by a strand (2) 0 otherwise

where nmK is the number of monomers per Kuhn segment. In the above expression, Θ is the Heaviside function. The local strand segment density distributions F(R BB j)/2, |R Bj - R B i|, nKij) are measured in Kuhn (R Bi + R statistical units (i.e. units that follow a random walk statistics) per unit volume. They are conveniently expressed in terms of a reduced strand density F˜ as F ) nKijF˜ /lK3. F is the density (in Kuhn segments per unit volume) in the region specified by the position vector R B , due to a strand of length nKij Kuhn segments, given that the start of the Kuhn segment 1 and the end of B i and R B j, the Kuhn segment nKij are fixed at positions R respectively (see Figure 3). Local Density of Segments. In eq 1, the cell density FW is expressed in terms of a reduced segment density distribution F˜ of a strand tethered at both ends. Fischel et al.8 have derived the exact solution for the end-toend distance distribution of freely jointed chain strands and applied it to estimate the distributions of the intermediate segments of the strands. Although the exact solution is known, its implementation in the problem studied here is computationally very expensive. For some regions (for example, short distance between the nodes and large number of Kuhn segments), it is

usually preferable to use approximate solutions (i.e., the corresponding solution for a Gaussian chain strand), which are less accurate (usually with few percent error) but much less expensive computationally. In the present work, we perform a numerical (very accurate) calculation of the local density distributions of tethered freely jointed chains. In this calculation, a tethered strand is represented as a sequence of bonds, the length of each bond being equal to the size of a Kuhn statistical unit. At fixed spatial distance between the two terminal nodes and with known number of Kuhn segments in the strand, we use Monte Carlo sampling in order numerically to compute and tabulate the segment density. We start the Monte Carlo sampling with an equilateral V shape for the strands (each side of the V connecting a tethered end to a point on the perpendicular bisector of the line segment connecting the tethered ends). “Pivot” MC moves are used,9,10 which entail rotations of randomly chosen parts of the strand about the axis connecting their terminal segments. The rotation angle is chosen randomly from 0 to 2π. The pretabulation of the density is performed for a strand with number of Kuhn segments determined by the average molecular weight between entanglements (i.e., 34 Kuhn segments). The values used for the end to end distance ranged from zero to full extension, with a constant interval between successive values; results were pretabulated for a total of 101 end-to-end distances. For intermediate values we have proved that a linear interpolation is an excellent approximation. Because of symmetry of the distribution of tethered chain segments about the axis of the end-to-end vector, only two spatial coordinates are necessary in the pretabulation: z, along to the end-to-end vector, and y, normal to it (see Figure 3). To find the density of a strand at position R B in space, with its first segment B j, the beginning at R B i and its last segment ending at R corresponding z and y values can be determined using the dot (‚) and cross (×) products of vectors (where the origin for z and y coordinates is the position of the midpoint of the distance between the tethered ends):

(

R B-

z)

y)

(

B|R

)

R Bi + R Bj ‚(R Bj - R B i) 2 |R Bj - R B i|

(3)

)

R Bi + R Bj × (R Bj - R B i)| 2 |R Bj - R B i|

Moreover, we have shown that, for any number of segments, a Gaussian scaling of the density with respect to the distance between the tethered ends is valid for

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extensions up to some “upper limit” value, which depends on the number of segments. For large number of segments in the strand and small distance between the tethered ends, this Gaussian scaling is confirmed analytically.8,11 A linear scaling of the density with respect to the distance between the tethered ends is valid for extensions larger than this “upper limit” value. Actually, for a given number of segments at extension exactly corresponding to the “upper limit” value, Gaussian and linear scaling are equally well applicable. By linear scaling we mean that the reduced strand density F˜ at position (z, y) of a tethered strand of na segments with ends at spatial distance R12a from each other is the same as F˜ of a strand of nb segments with ends at spatial distance R12b ) (nb/na)R12a at position [(nb/na)z, (nb/na)y]. By Gaussian scaling, on the other hand, we mean that F˜ (z, y) of a strand of na segments with ends a distance R12a apart is the same as F˜ (z, y) of a strand of nb segments with spatial distance between the ends R12b ) x(nb/na) R12a at position (x(nb/na)z, x(nb/na)y). Our studies have shown that special care is needed for strands consisting of a very small number of segments. In this case, however, as we have to distribute in space a very small number of segments, the results are practically very similar, no matter what scaling is adopted. In our calculations, it is the reduced strand density F˜ that is pretabulated as a function of scaled z and y coordinates, with parameter a nondimensionalized strand end-to-end distance R:

F˜

(

)

|R Bj - R B i| y z , ,R≡ h h lKnKij lKnKij lKnKijh

Here h ) 1 for linear scaling and h ) 1/2 for Gaussian scaling and |R Bj - R B i|/(lKnKij) takes values from 0.00 to 1.00 with step 0.01. The local Kuhn segment density at R B due to the strand ij is calculated immediately, once the reduced strand density is known:

(

FR B-

) (

)

nKij R Bi + R Bj z y ,|R Bj - R B i|,nKij ) F˜ , ,R 2 lKnKijh lKnKijh lK3 (4)

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between segments in the same cell (AWattr) and also between segments in adjacent cells (AW,Vattr). Elastic Interactions. The elastic force between connected nodes arises due to the retractive force acting to resist the stretching of a strand. This force originates in the decrease in entropy of a stretched polymer strand. The Gaussian approximation can be used for most extensions.8 In this approximation, the force on node i due to its connection to node j is12

F Bijentropy string ≡ - ∇RB ijAijentropy spring ) 3kBT

R B ji nKijlK2

(6)

where R B ij ≡ R Bi - R B j. For higher extensions the inverse Langevin approximation or an even more accurate functional dependence, computed by simulation, could be used. However, as the contribution of the elastic energy to the total free energy is minor, such computationally more expensive expressions were not implemented in the present study. The elastic force depends on the coordinates of the nodes, but it does not depend on the local network density. The corresponding expression for the free energy of an entropy spring is

Aijentropic spring )

(R B ij‚R B ij) 3 kBT 2 n l 2

(7)

Kij K

Repulsive Interactions. The repulsive force ensures that the density distribution is such that segments do not overlap. The simplest approximation is to identify each segment with a hard sphere. The best known equation of state of the hard-sphere fluid is the Carnahan-Starling equation of state, βP/F ) (1 + η + η2 η3)/(1 - η)3, with β ≡ 1/kBT, P being the pressure, F the number density, and η the packing fraction. This equation of state provides an excellent fit to the results of computer simulations over the entire fluid range. From this expression, a simple form is obtained for the Helmholtz free energy in cell W:

(4ηW - 3ηW2) βAWrep 3 ) ln(FWΛ ) - 1 + N (1 - η )2

(8)

W

Free Energy Calculation. The free energy of the network has three contributions: (1) the elastic energy of the stretched strands; (2) the attractive interactions between segments; and (3) the repulsive interactions between segments:

A)

∑ i