# G Zaikov - Degradation of polymers in aggressive media - страница 1

CHEMISTRY & CHEMICAL TECHNOLOGY

Vol. 3, No. 1, 2009 Chemistry

Gennady Zaikov and Marina Artsis

DEGRADATION OF POLYMERS IN AGGRESSIVE MEDIA.

KINETIC APPROACH

N.M. Emanuel Institute of Biochemical Physics, Russian Academy of Sciences 4 Kosygin str., 119334 Moscow, Russia chemb io @sky. chph.ras.ru

Received: September 01, 2008

© Zaikov G., Artsis M., 2009

Abstract. The degradation of polymers in aggressive media is a complex physico-chemical process including adsorption, diffusion and the dissociation of chemically unstable bonds. The course of degradation has a number of special features, which are linked both with the specific structure of polymeric materials and with specific kinetics of reactions in solids.

Key words: degradation, stabilization, aggressive media, carbochain and heterochain polymers, chemical kinetics.

1. Fundamental Kinetic Equations

In the course of degradation of polymers in aggressive media there are the following fundamental processes:

1) adsorption of the aggressive medium on the surface of the polymer article;

2) diffusion of the aggressive medium within the polymer article;

3) chemical reaction of the aggressive medium with chemically unstable groups of the polymer;

4) diffusion of degradation products to the surface of the polymer article;

5) desorption of degradation products from the surface of the polymer article [1-10].

A mathematical consideration of these processes is very difficult, and, accordingly, it is usually assumed that one or at most two stages are slow in comparison with the others, and consequently limit the course of the whole degradation process. The first and the fifth stages usually take place more rapidly than the second, the third and the fourth ones.

The diffusion of degradation products to the surface of the polymer article usually plays insignificant role in the assessment of the change in the service properties of polymer articles in the course of degradation, but it may restrict, for instance, the process of biodegradation of a polymer within a living organism. In the course of chemical degradation the polymer and the aggressive medium (liquid or gas) are in separate phases. The chemical reaction between the aggressive medium and chemically unstable groups in the polymer may take place either at the interface phase or within the polymer phase (components of the aggressive medium which are dissolved in the polymer take part in the reaction).

Regarding these two phases as a single closed system, the rate of the chemical reaction w may be expressed as

dn

, where n is the number of groups which have decomposed at time t, and V is the volume of the polymer.

S

In the first case w » y, where S is the surface area

of the polymer article and, in the second case, this relationship is more complex. Let us consider it in details. Assuming that the law of the mass action is observed, the volume of the polymer remains practically unchanged during the course of degradation and the polymer is isotropic in properties; then we can write the equation for the rate of decomposition of the chemically unstable groups as follows:

where c° is the initial concentration of chemically unstable groups in the polymer; Cn is the concentration of decomposed groups; CCat is the concentration of the catalyst in the polymer; Csolv is the concentration of the solvent in the polymer; k is the rate constant of decomposition of the unstable groups.

The concentration of the catalyst in the polymer, e.g. acid or base, can be found from the equation (2).

dCcat = D V2C -YC Ck (2)

^ ^CCit^ '"Cat /_4l-Caf-i*,i (2)

where V is the Laplace operator; Ci is the concentration of functional groups in the polymer which are capable of taking part in a complex formation or substitution reaction;

k. is the rate constant of the complex formation or substitution reaction of the catalyst with functional groups of the polymer.

The second term on the right-hand side of Eq. (2) takes into account the possibility of such reactions as protonation, interaction with a hydroxide ion, and so forth.

If the solvent is involved in decomposition of chemically unstable groups in the polymer, for instance the removal of water in hydrolysis reactions, then the solvent concentration may be found from the equation (3).

dcsoiv — D V2r — k(c0 - c V c (3)

While writing this equation it is necessary to justify a number of assumptions.

First, the polymer-aggressive medium system is, as a rule, very much diluted in relation to the aggressive medium, i.e. it may be assumed that D and D , do not

7 J cat solv

depend on the concentration of the corresponding components in the polymer. If the aggressive medium is soluble in the polymer to any considerable extent, it becomes necessary to use an equation which takes into account the relationship between D and the diffusant concentration.

Second, the value of k is constant. This is correct when the aggressive medium has a low solubility in the polymer. In general, the influence of the aggressive medium on k can be reduced, as a first approximation, to the influence of the solvent on the rate of the chemical reaction, which can be predicted on the basis of existing theories [11].

Third, the decomposition reaction of chemically unstable groups is practically irreversible. This condition is realized when degradation takes place under some particular reaction conditions to quite low degrees of dissociation (< 0.05). In this case the concentration of chemically unstable groups changes only slightly and c° - с » c°.

n n n

Thus, determining the rate of decomposition of chemically unstable groups in a polymer under the action of aggressive media means the joint solution of Eqs. (1)-(3).

Usually the following boundary conditions are assumed:

- the concentration of substances diffusing to the surface of a polymer article is a constant value (in practice this condition is fulfilled with a rapid flow over the article of a stream of solution with the constant concentration of the aggressive medium);

- the concentration of substances diffusing to the surface of the polymer article is a function of time, and the dependence of the concentration on time is expressed by one of the adsorption equations;

- the concentration of substances diffusing to the surface of the polymer article is determined by some particular law of mass transfer, the simplest being j — b(csuif — cp ), where j is the flow of substance diffusing; В is the mass transfer coefficient; c , and c

'->7 і 7 surf p

are the concentration of the substance on the surface and within the volume of the solution of the aggressive medium, respectively.

The majority of polymer articles may, to a first approximation, be regarded as simple geometrical bodies.

We can take a parallelepiped as a model of polymer films and coatings and a cylinder as a model of a filament and so forth.

We shall now consider the solution of Eqs. (1)-(3) for a parallelepiped and a cylinder.

2. Macrokinetics of Chemical Degradation

The process of degradation may take place in various regions, depending on the ratio of rates of the diffusion process and of the chemical reaction.

The rate of diffusion of the aggressive medium is commensurate with the rate of the chemical reaction, and degradation takes place in a particular reaction zone which size increases with time and ultimately reaches the limits of the dimensions of the polymer article, .e. the reaction takes place under internal diffusion-kinetic conditions.

The rate of diffusion of the aggressive medium considerably exceeds that of the chemical reaction. After dissolution of the aggressive medium in the polymer is finished, the degradation takes place over the total volume of the polymer, .e. under internal kinetic conditions.

The rate of diffusion of the aggressive medium is considerably lower than that of the chemical reaction. In this case degradation takes place in a particular thin reaction surface layer or, as it is usually put, from the surface of the polymer article, .e.,under external diffusion-kinetic conditions.

2.1. Internal Diffusion-Kinetic Conditions

The joint solution of Eqs. (1) - (3), even with the above assumptions, is a difficult mathematical problem and is possible only by using a computer. While considering the diffusion of the aggressive media in polymers, it has been established that the diffusion of the catalyst (acid or base) and of the solvent takes place at an identical rate on a single front and that the diffusion of the solvent is at a considerably higher rate than that of the catalyst.

The first variant occurs, as a rule, in the diffusion of the aggressive media in hydrophilic polymers and in the diffusion of acids and bases with a high vapor pressure in hydrophobic polymers.

The second variant occurs in the diffusion of acids and bases with a low vapor pressure in hydrophobic polymers.

These patterns of behavior simplify the solution of the problem, since, in the first case, the diffusion of the catalyst and solvent can be characterized by a single diffusion coefficient D = D ,. In the second case, D ,

cat solv 7 solv

>> D , and it can be reckoned that the concentration of

cat

the solvent in the reaction zone of the polymer article becomes constant after a certain time and equals to its solubility C°, , i.e.

dC

solv

dt

0 when Csolv = C.°olv .

Thus, the problem is simplified and is reduced to the solving of Eqs. (1) and (2), of which, taking into account the relationship Cn0 - Cn Cn0 , the second takes the form

dt keffCCat (4)

where

k

eff

kC0

solv

In Eq. (2), for simplicity, we restrict ourselves to one term and make c = c.

Depending on whether or not the combination of the catalyst with functional groups of the polymer takes place, we may obtain two equations for the change in the number of macromolecular scissions in the course of degradation.

K_®_2°, i.e. the functional groups present in the polymer formed in the course of degradation react with the catalyst practically irreversibly. Bearing in mind that these reactions (e.g., protonation or complex formation) proceed far more rapidly than degradation reactions, Eq. (2) takes the form

dC

CCt = D V 2C - k C

(5)

The solution of this equation was first achieved by Danckwerts [12], on the basis of the solution of the analogous problem of thermal conductivity considered by Carslaw and Jaeger [13]. The solution of Eq. (5) for a parallelepiped one of which dimensions (l) is much less than the other two with the boundary condition CCat - CC0at with x = 0 and l with t ® 0 and the initial condition C = 0 with t = 0 and 0 £ x £ l takes the form

CCat (X t) - CCat X

\1 - - Y7-^-X [kJf + b*D t exp-(b2D t + kff >u (6)

[ к m-0 (2m + ijblDaat + 1)L eff m Ca,V Xm Cat efJfn\

where bm--1-; C°at is the solubility of the catalyst

in the polymer.

Substituting Eq. (6) into (4), and carrying out double integration from 0 to t and from 0 to l/2, we get

Cn - ke CC0attX

j 1 - _8_ Y kef (b2mDa + kef)+ bmDCat [1 - exp- (bmDCat + kef )]} (7) 1 к 2 Y (2m + 1)bm (bm,Dcat + kef j2 lt

Examination of Eq. (7) shows that if bmDCCtt > kef, it is sufficient to limit oneself to the first term of the series. A detailed consideration of this case will be given below.

Solution of Eq. (5) for a cylinder of radius r much less than the length l with the boundary conditions C t — C° t with x = r with t > 0 and the initial condition

C t C t

0 with t = 0 and x £ r, takes the form

C =

C a t

CCCt (x,t) — CL X

1 -2 Y

keff + DCat ^ exp-|D,

CC^t + keff lt

J 0| x

r

DCal— + keff J1\mn)

r J r

(8)

where J_ and J1 are Bessel functions of the first kind of zero and first order, respectively; and mn are the roots of the Bessel function.

An expression for Cn is obtained from Eqs. (4) and (8) after integration from 0 to t and from 0 to r:

Cn — kefC0at Y

D

r z

1 - exp (- zt)~|

(9)

where z

D

— + k

2 + kef

For the initial time of degradation, if exp (-zt) > 0.9, we can use the approximated equation (10) for calculations:

Cn » effCCat [ 3к"2 r

2r2

2

'

DCC2 Г D

e

(10)

If exp (-zt) < 0.1, then Eq. (9) takes the form

Cn » kefClat Y S2 і t n—1Zr V

Dcat ml

If —ca 2 " > kef, then, as in the case considered

above, it is sufficient to restrict ourselves to the first term of the series, which simplifies Eqs. (9) and (11).

K_ ® 0, i.e., the functional groups in the polymers do not react appreciably with the catalyst. Eq. (2) takes the form of the Pick diffusion equation, the solution of which, for the parallelepiped (film) and cylinder (filament) mentioned above, is discussed in literature.

On substituting the solution into the Fick equation for the parallelepiped in (4) and integrating within the same limits as in Eq. (9), we get

2

n—1

z

n—1

5/2

r

r

X

m—0

Cn — ke„C.t

- £1[1 - exp(2m+1)2 y](2mmhry\ (12)

where

K 2

l2

With y < 1, this case is realized in the initial period of degradation for films of any particular thickness

Cn — keffClatt Л j(y)

к

(13)

where <P (y ) —Y

exp- (2m +1)2 y -1 + (2m +1)2

y

m—0 (2m +1)4 y

By using a computer there is established a simple relationship j(y)— 0.589y1/2, i.e.,

4

C —_k C0 D1/2l"V32

n 1/2 eff cat1^ cat1 1

к

(14)

Eq. (14) is conveniently expressed as follows:

8

n —

к

1/2 "eff^ca^cat

(15)

Thus, in the initial period of degradation the number of groups which have decomposed depends on the rate constant of the chemical reaction, on the diffusion coefficient, on the solubility of the catalyst, on the surface area of the film and on time.

## Похожие статьи

G Zaikov - Degradation of polymers in aggressive media

G Zaikov - Alkylation of 26-di-tert-butylphenol with methyl acrylate catalyzed by potassium

G Zaikov - Biodegradation of polymeric materials generalized kinetic data