Description of Polymerization Dynamics by Using Population Density

10 Description of Polymerization Dynamics by Using Population Density Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 13, 2017 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch010

D. C. TIMM and J. W. RACHOW Department of Chemical Engineering, University of Nebraska, Lincoln, Neb. 68508

Molecular distributions for chain-growth polymerizations are described in terms of a model based on conservation of popula tion, a lumped parameter model. In the area of experimental analysis, theoretical relationships readily yield to grapical inter pretation. Steady-state macromolecular distributions for poly styrene as functions of degree of polymenzation, initiated by n-butyllithium and by 2,2-azobisisobutyronitrile, correlate. Po lymerization dynamics are simulated by using the method of characteristics. Transient response surfaces for an anionic, batch polymerization of styrene are such that a single integration is required. Step responses to perturbations in residence time are simulated for a free radical polymerization of vinylidene chloride in a continuous, well mixed tank reactor.

S

tandard procedures for simulating molecular distributions, including num ber and mass fractions, as functions of degree of polymerization and time are quasi-steady-state approximation (1, 2, 3), Eigenzeit transformation (4), Ζ transform (5, 6, 7), generating functions (8, 9), and direct integration (10, 11). The intricacy of relationships is such that the implementation of molecular distribution data explicitly into kinetic analysis has yet to be developed. Pub lished work (12, 13, 14) demonstrates the complexity of simulating molecular distributions given kinetic rate information a priori. Experimental techniques for updating kinetic mechanism subject to molar or mass distributions of resin ous materials have not been reported. The major objective of this research is to develop an alternative analysis, one which is amenable to interpretation of experimental data as well as numeri cal simulations. Population density analysis results in working expressions that yield to graphical interpretation, including free radical and carbanion polymeri zation mechanisms. Experimental verification is included. Dynamic response surfaces for batch and continuous modes of reactor operation are simulated by using the method of characteristics. Molecular Distributions Resinous materials are comprised of macromolecules distributed according to degree of polymerization. Traditionally, the distribution is discrete, being defined only for integer values of degree of polymerization. As shown later, 122

Hulburt; Chemical Reaction Engineering—II Advances in Chemistry; American Chemical Society: Washington, DC, 1974.

10.

τίΜΜ AND RACHOW

123

Polymerization Dynamics

distinct advantages exist for a description in terms of a continuous independent variable. Population density is defined implicitly by the integral ~

n

(1)

A(n,t) dn = N(k,t) - N(j,t)

k

where the continuous variables A(n,t) and N(k,t) are the differential, molar distribution and the integral, molar distribution, respectively. Population density is defined explicitly by (2)

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 13, 2017 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch010

A(n,t) = dN(n,t)/dn

Thus, population density is the slope of the continuous cumulative molar dis tribution, evaluated at a specific degree of polymerization and time. The physical significance of the differential distribution, Equation 2, and the cumu lative distribution, Equation 1, will be exemplified in the following derivations. A Carbanion Polymerization. Styrene will polymerize by the following carbanion mechanism (15, 16, 17) if initiated by n-butyllithium : Initiation I At

Κ

Α

+ Αι


10.

τίΜΜ A N D RACHOW

125

Folymenzation Dynamics

In the limit, A (1,0 = Κι I{t)/K

(5)

p


Molar conservation equations of monomer and initiator define these timedependent coefficients:

jjga _

*,(o - * ( ο _

dhj^f)

/. ,i(t) - I (t)

=

exp

Tp

Κ

ρ

Μ

_

{

R

ι

i

)

Α

I ( f )

τ

ο

M

τ

{

(

t

ι

)

( 6 )

)

( 7 )

The total, cumulative molar concentration of unassociated polystyryl anions is derived by integrating Equation 4 with respect to degree of polymerization: f

œ

dA(n,t) ,

.

r

v

œ

dA(n,t) ,

= l f Ai(n,t) dn + k f" C(n,t) dn θJ l Jo This equation may be simplified by applying the Leibnitz rule to the first term, integrating the second term, and subsequently utilizing Equation 5 to eliminate A(l,t). For a convergent solution Α ( ο ο , ί ) = 0. The resultant expression is œ

j Ατοτ(0 t

r

3

= Κι 7(0 M(t) - [ ^ + kt Α ο τ ( θ ] Α ο τ ( 0 Τ

Τ

(8) A (n,t)dn + 2k CTOT(0

+ I

t

r

f

00

The total molar concentration of all associated complexes is C (t) = Vz J C(n,t)dn. The factor % is necessary because this surface is symmetric, A,; — A = A — Aj, and unique reactions only are required. The population density of associated polystyryl anions C(n,t) which con tain one molecule of specified degree of polymer may be derived by procedures analogous to that leading to Equation 4. The resultant expression is T0T

k

k

dC{n,t) + Qj+ fc) C(n,t) dt

+ k ATOT(0 A(n,t) t

(9)

In experimental investigations, the reaction is quenched with water or alcohol, resulting in the annihilation of unassociated and associated polystyryl anions. The resultant molecular distribution, determined by gel permeation chromatog raphy, will be comprised of macromolecules originating from associated com plexes and from free anions. Thus, the measured population density is T(n,t) « A(n,t) + C(n,0 by Equations 4 and 9 +

K

v

M(t) * £ 0

+ J T n,t) = ) Τ,ΟΜ) (


126

C H E M I C A L REACTION ENGINEERING

If polystyryl anion association phenomenon is at equilibrium (15, C(n,t) = K

II

16):

A (t) A(n,t)

eq

TOT

and T(n,t) = [1 + K

eq

ATOT(01 A(n,t)

thus Ττθτ(0


and dT(n,t) dt

(10)

= [1 + Keq -Ατθτ(01 ^TOT

+

Equations 4-11 define polymerization dynamics in a well mixed tank reactor, subject to stated assumptions and constraints. The utility of the model will now be demonstrated. STEADY STATE EVALUATION. Steady state, isothermal, continuous polymeri zations were used experimentally to test proposed methodology. Only monomer, solvent, and initiator were fed to the reactor, so Τ (η,ί) = 0. The steady state population density distribution obtained from Equation 11 is simply: {

T(n,ss) - T(l,ss) exp j -

j^-

r(ss) M(ss) i4 oT(ss)

(12)

T

100 Figure 1.

300

n-1

Steady-state molar distribution, n-butyllithium initiator

Figure 1 is representative of experimental distributions. Thus, there is excellent agreement between theoretical and experimental distributions. Procedures for determining population density T(n,ss) and T (t) from gel permeation T0T


10.

τίΜΜ AND RACHOW

127



•8r

T

TOT

Ό 01 02 (ss)/(eM(ss) (-slope ))

Figure 2.

Determination of rate constants K and Keq

P

chromatography are detailed by Rachow (22). The rate constants associated with propagation and dimeric associa tion, K and Kgq, respectively, may now be evaluated, utilizing (—slope) = p

^ ^ ^ — τ — r The latter may be solved explicitly for A 0K M(ss) ATOTÎSS) substituted into Equation 10. After linearization, 0 T

S

T 0 T

( s s ) and

p

0M(ss) (-slope) - ± K

p

r(ss) +ψ\ ^ K θ M(ss) (-slope) 2

(13)

p

Experimental data are presented in Figu/e 2, substantiating the procedure. The intercept T(l,ss) and initiator concentration provide data for the evaluation of the initiation mechanism. Specifically, T(l,ss) =

Κι 7 QT(SS) /(SS) , T

Κ ρ ATOT(SS)

Experimentally determined n-butyllithium concentrations are the sum of the unassociated and associated molecules. Iexp = y [Iy] + A ο If the initiator is predominantly associated, A < y [ i ] , then 0

y


128

C H E M I C A L REACTION ENGINEERING

1 TOT(SS)

II

L y J

Experimental data at 5 0 ° F are presented in Figure 3. The association number y is nearly 3. Research in progress will exemplify these preliminary observations. However, they do demonstrate a practical experimental application which incorporates molecular distribution analysis into a kinetic evaluation. The simplicity of working expressions is self-evident.


B A T C H REACTOR DYNAMICS.

Published descriptions of differential, molar

distributions are complex, nonlinear functions. The following relationship has been derived (23), utilizing a discrete, degree of polymerization model: K

P

d

T2

V ^ T O T ( ^ 2 ) 2K

e

VÎTTOT(TÎ)/

\V2KZJ

XI

/

where the Eigenzeit transformation is dr = M(t)dt. Thus to describe the molecular surface, numerous integrations at specific degrees of polymerization must be performed. The significance of the factorial and power can only be appreciated with the realization that the degree of polymerization ; exceeds many thousands for commercial grades of polystyrene. The population density surface, on the other hand, may be evaluated by exact solutions coupled with a single, simple integration (24). Subject to the operational constraint of a batch reactor and the Eigenzeit transformation, Equation 11 simplifies to * y + i + * ; M * y - ° θτ 1 H- A A TOT (τ) θτ The initial macromolecular distribution is T(n,0). If experimental conditions yield predominately unassociated initiator, the boundary condition, Equation 5, with the aid of Equations 7 and 10 is simply g

(

i

5

)

e q

Γ ( 1

' > τ

=

11 ++ κ" A

4 A. TOT ( τ ) Ρ ( 0 )

e q

β Χ

(

"

K

l

T )

(

1

6

)

Solving the partial differential equation by the method of characteristics (25, 26), the following relationships describe the population density surface:

dT(n,x) ds dn ds

1 + A~

eq

=

0

( 1 8 )

Κ A TOT ( τ )

(19)

The dependent variable A ( T ) may be evaluated in terms of Τ τ ο ( τ ) , Equation 10. Therefore, Equation 15 is initially integrated with respect to degree of polymerization and Equation 16 is used to eliminate Τ ( 1 , τ ) , yielding: t o t

Τ

Ττοτ (τ) = T O T ( 0 ) + 7(0) [1 T

exp (-

Κιτ)]


(20)


10.

τ ί Μ Μ A N D RACHOW

129


logdexp)

Figure 3.

Determination of Ki(K /y) a

1/v

Utilizing Equation 10: Λ

ΛΤΟΤ

( ^ (τ;

=

- 1+ V l+

e q

Ττοτ (τ)

, * 9

κ*ι)

4ly eq

For a characteristic emanating from the initial condition plane, suitable initial conditions for Equations 17-19 are s = 0, τ = 0, T(1,0) (see Figure 4). The characteristic equation maintains a constant magnitude Γ (1,0); its reflec tion onto a constant Τ ( η , τ ) = 0 plane is determined by integrating Equation 19, utilizing dr = ds. The remainder of the surface can now be generated without additional integration by using explicit relationships. Each trajectory emanating from the initial condition plane will be of constant height (Equation 18), as specified by the initial macromolecular distribution Τ ( η , , 0 ) . Their reflections onto the plane Τ ( η , τ ) = 0 will remain a constant distance, η ( τ ) + n = η ( τ ) , from the previously generated η ( τ ) curve. Sufficient trajectories are rapidly gen erated to define that portion of the surface that results from the initial macromolecular distribution of polystyryl anions. The population density surface resulting from initiation kinetics is gen erated by a similar procedure. Trajectories emanating from the boundary con dition plane have magnitudes Τ ( Ι , τ , ) as determined by Equation 16. Their reflections onto the plane Τ ( η , τ ) = 0 will again be of constant distance from ;

;


130

CHEMICAL REACTION ENGINEERING

II

the originally generated η ( τ ) curve. Sufficient trajectories to describe the sur face are generated. Intermediate points may then be determined by interpo lation. For the case y 1, numerical integration will yield the boundary condition Τ(1,τ) and the total molar concentration Τ (τ).


τ ο τ

Figure 4.

Response surface, living polymer

Free Radical Polymerizations Population density will now be shown to be an effective variable for simulating and for experimental interpretation of free radical polymerizations. A representative mechanism is: Initiation

/ -»2A

Κι

0

Propagation

Ai + M

Transfer solvent

Ai + S^Pi

monomer Termination disproportionation combination

—>

Aj+1

Ai + M -*Pi

κ*

+A +A 0

0

Kt

m

Ai + A* -> Pi + P

Kd

Ai + A -> P

Ktc

k

k

j+k

t

The reactivity of radical fragments generated in transfer reactions is assumed to be equal to that of primary radicals formed by initiation kinetics. Through basic conservation of population laws, the following relationships may be developed:


10.

τίΜΜ AND RACHOW

a^pO

+

131

Polymerization Dynamics sA^t)

K p M ( i )

J1

+

+

[

K

t

d +

R t c ]

Α

ύ

ο

ύ

{

1

)

(22) + Kts S(t) + X

t m

Μ ( θ | A(n,i) =

^j^-

Λ /π A 2Ki//(Q + {KfSg) + g M ( 0 } ATOTW A (0,0 -

s (23)

- [K, + K ] Mit) ΑτοτίΟ

(24)

to


«

-

;

M

i

i

)

m

tm

jgO . / . ( Ο - / ( Q at

αΆτοτ(0 + I AtotU) eft οι

(

2

5

2

6

)

)

0

+ [Κ* + K ] [ATOTW]' = f*AAn,t)

d

ti

θ

V

+

n

' '' )

2

Α

/(ί

(

JX-0 (27)

+ [fftd Α ο τ ( 0 + #ts 5(0 + Ktm Μ(0] Α(η,0; η > 0 Τ

Χ

00

A(n,t)dn. The term corresponding to the rate of termination by combination in Equation 27 includes all distinct, alternative reactions by which two smaller macroradicals may couple to form a single, cessated polymer molecule. Steady State Polymerization of Styrene. Steady state distributions of macroradicals and terminated polymer molecules is readily obtained from Equations 22 and 27. If Α ^ η , ί ) = T^nj) = 0 for polymer-free feed, then \ \ + [tftd + tftJ A TOT(SS) + K Siss) + K A in,s) = A (0,ss) exp i — ^ ( K Miss) ts

Miss)

tm

θ

p

Pgg

_

θ

jnX A(0, ) TC

SS

+

κ

^

Α

τ

ο

τ

(

8

8

)

+

S

(

M

) +

K

t

m M

(

s

g

)

J

[ η Γ (28) )

(

2

9

)

The experimentally measured distribution will be comprised of macroradicals and terminated polymer molecules: T(n,0 = Ain,t) + P(n,0

(30)

For polystyrene initiated with 2,2-azobisisobutylronitrile, the following numbers are representative of kinetic parameters and steady-state concentrations (12). = 1.4 X 10~ /sec

Miss) =2.4 molar

K

= 4.4 Χ 10 liter/mole sec

iS(ss) =7.4 molar

K«

= 2.9 Χ 10" liter/mole sec

J(ss) = 0.0152 molar

4

P

2

3

Ktm = 3.2 Χ ΙΟ" liter/mole sec 2

K

= 1.2 Χ 10 liter/mole sec

Kd

= 0

8

tc

t

temp. = 80°C θ = 3600 sec


132

CHEMICAL REACTION ENGINEERING

II

-12c-


-13 CO to

CD

o -14

-15. 'Ο

400

Figure 5.

.ΐ-

1200

800

η

Molecular distribution, AIBN initiator, styrene

Therefore, 1 = 0.0003; [Ktd + K ] ATOT(SS) = 15.2; K S (ss) - 0.02; θ t e

tc

K

t m

M(ss) = 0.08; at η - 10,

η Α

^Α(0,

8 8

) .

L

1

Thus for η > 10, m, χ 4/ χ Γι ι 0 K Γ(η, ) = Λ(0,«) [1 + Λ

Β

t e

4(0,ss)nl J

e

x

p

/ ("

K

A oT(ss)n\ g M( ) j t c

T

p

S S

A good approximation is, if η > 10, T(n,ss)

0 K «4 (0,ss) t

2

exp

-

Ktc ^TQT(SS) η I K M(ss)

(31)

p

Thus, population density analysis provides a relationship which readily yields to graphical interpretation and which is a much simpler relationship than a Flory-Schultz description. Experimental data (27) are summarized in Figure 5, subject to the constraint of neglecting population densities of oligomers. To evaluate kinetic constants, the additional relationships of measurable quantities are: Population distribution of macromoleeules

( — slope) = ^ j ^ ^ g ^ intercept = θ K

t c

4 (0,ss)/2 2


10.

τ ί Μ Μ A N D RACHOW

133


2Ki /(ss)/ Κ Ρ M (se)

Initiation kinetics

A(0,ss) «

Styrene concentration

Min(ss) — M (es) » # An M(ss) θ

Macroradical concentration

XpM(ss) A(0,ss) = KtcA oT(ss)

D

T

r(ss) 2


This set of five equations provides a means for the determination of five unknowns (three rate constants, population density of primary free radicals,

-7r

-8

0

—

100 200

c

Description of Polymerization Dynamics by Using Population Density

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