C OMPOSITION OF C OPLYMER WITH N M ONOMERS FROM S TATE S PACE R EPRESENTATION

The stability of the system of reactions can be studied by obtaining the Eigenvalues of the K matrix. The eigenvalues of the K matrix are obtained from the roots of the characteristic

There is significant interest in obtaining copolymer product via free radical polymerization reactions from more than one starting monomers. SAN, styrene/acrylonitrile copolymer is an example of product manufactured profitably with two starting comonomers, styrene and acrylonitrile. Terpolymers are those with 3 starting monomers. Tetrapolymers are product made from four starting monomers getting into the polymer. Consider n monomer repeat units entering a polymer backbone chain in a CSTR [Sharma, 2012]. There are n² propagation reactions that can be involved in the multi-component copolymerization. Thus for tetra polymerization there would be 16 propagation reactions involved. These reactions are listed

Mathematical Process Models 145

The rate of irreversible reactions can be written as;

     

The free radical species formed may be assumed to be highly reactive and can assumed to be consumed as rapidly as formed. This is referred to as the quasi-steady state assumption, QSSA. Thus;

   

Mathematical Process Models 147 concentrations by represented by vector M* as;

1

The multicomponent copolymerization rate constants can be represented by a rate matrix

The rate matrix for multi-component copolymerization can thus be also be expressed in terms of the homopolymerization propagation rate constants of the n monomers respectively and reactivity ratios as defined in a similar manner to Eq. (3.264). The set of

Only the diagonal elements of the resulting matrix from the RHS, right hand side of Eq.

(3.262) is of interest. The rate equations that represent the radical generation and consumption during propagation can be given in the matrix form as;

* ( * )

)

d

dt   K



~

M MM K(M*M

Eqs. (3.267) and (3.268) form a set of autonomous systems. For certain values of the rate propagation matrix using Eigenvectors and stability analysis it can be shown that periodic solutions may result. The initial conditions of the monomers can be represented by;

Mathematical Process Models 149

M (t=0) = M0 (3.269)

By the QSSA, quasi-steady state assumption, i.e., the radical species are highly reactive, Eq. (3.269) can be set to zero. Or,

( * )^T  )

KT MM K(M*M ^T (3.270)

For a given set of reactivity ratios, homopolymer propagation constants and initial condition of multi-component comonomers, M* can be solved for using Eq. (3.270) and then substituted in Eq. (3.267). Then the monomer concentrations with time as well as the polymer composition can be calculated from Eq. (3.270). Observing Eq. (3.270) can be solved for by the method of Eigenvectors [Varma and Morbidelli, 1997]. It can be seen that non-trivial equations to the set of ordinary differential equations with constant coefficients represented by Eq. (3.267) exist only for certain specific values of  called Eigenvalues. These are the solutions to;

 I-K  0

Eq. (3.271) upon expansion may lead to a polynomial degree of n in;

Pn() = 0 (3.272)

In order to obtain numerical values for  the expansion of Eq. (3.272) is considered obtained by Laplace development of the determinant;

 

The copolymer composition can be obtained using the trace of the matrix defined in Eq.

(2.102). Trace is a sum of all the diagonal elements of a matrix. Thus;

1 ~

The corresponding monomer compositions in the CSTR for n monomers entering the copolymer backbone chain can be written as;

1 obtain the relation between the multi-component copolymer composition and CSTR monomer compositions. It can be realized that trace of a square matrix is a scalar quantity.

Substituting Eq. (3.277) in the set of copolymerization rate equations the rate matrix can be rewritten as;

Eq. (3.277) is a reasonable assumption for multi-component copolymerization for the general case of n monomers as well. This implies that the rate of copolymerization propagation of MiMj

* equals the formation of MjMi

* radical. When this is applied to all possible pairs of monomers in n monomers the QSSA holds as the net production of radicals equals the consumption during the copolymerization propagation itself. Nothing about the termination reactions are brought forward into this analysis. Thus for n monomers,

*

Mathematical Process Models 151 The solution to Eq. (3.279) can be obtained by the method of Eigenvectors.

Assume that the solution to Eq. (3.279) has the form;

e t

M Z (3.280)

Where Z is the unknown vector of constants and  are the Eigenvalues. Substituting Eq.

(3.280) in Eq. (3.2782);

*

~ T

M KM Z (3.281)

Eq. (3.281) requires that e^t  0. Assuming Eq. (3.281) as the form of the solution to the set of ordinary differential equations with constant coefficients requires that Z  0. It can be seen from Eq. (3.281) that  is an Eigenvalue of the rate matrix modified with the free radical concentrations. The free radical concentrations can be obtained from Eq. (3.281). The modified rate matrix transposed, is a square matrix of nxn. Therefore n Eigenvalues can be expected.

Example Composition of Tetra polymer made in CSTR

Develop the monomer/polymer composition equations for a tetra polymer made from the monomers;

Eq. (3.278) can be solved for by the method of Eigenvectors [Varma and Morbidelli, 1997].

The reactivity ratios for the 4 monomer system considered are follows [Sharma [2012]};

r12 = 0.29; r21 = 0.02; r42 = 0.84; r24 = 1.5; r32 = 0.03; r23 = 0.14; r13 =1.2; r31 = 0.14;

The free radical concentrations can be solved for from;

)

Although there are 16 simultaneous algebraic equations generated from Eq. (3.285) it can be realized that M1M1 terms of the monomer compositions, M1, M2, M3 and M4 and using the homopolymerization propagation rate constants, k11, k22, k33 and k44. Eq. (3.282) can then be solved for in terms of the concentrations of the four monomers as a function of time. Four simultaneous equations and 4 unknowns, M1

*, M2

*, M3

* and M4

* can be obtained by multiplying the jth row and jth column on both sides of Eq. (3.285). i.e., the first row and first column, second row and second column, third row and third column and fourth row and fourth column. The (j,j) cell on both sides of the matrix equation can be seen to vanish. The equations are then;

* 2 3 4 * * *

Mathematical Process Models 153

* can be solved from the solution of the four equations Eq. (3.286-3.289). The composition of the tetra polymer can be obtained by obtaining the incremental rate of addition of the monomer of interest into the copolymer. These are given by Eqs. (3.275) and (3.276). The stability of the system of reactions can be studied by obtaining the eigenvalues of the K matrix in Eq. (3.283). Should the eigenvalues be all negative, then the system is stable. If all but one values are negative then it is an integrating system. Should the eigenvalues be positive it is an unstable system. Should the eigenvalues be complex conjugates then the system is considered to be oscillatory. This is expected to be subcritical and damped.