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Diffusion-controlled reactions in free radical polymerization—III
European Polymer Journal, 1965, Vol. I, pp. 247-252.
Pergamon Press Ltd.
DIFFUSION-CONTROLLED RADICAL
Printed in England.
REACTIONS
POLYMERIZATION--Ill.
IN FREE
AN ESTIMATE OF THE
IMPORTANCE OF GEMINATE RECOMBINATION OF RADICALS(1) C. R. PATmCI¢ and P. E. M. ALLEN Department of Chemistry, University, Birmingham, 15, England Department of Physical Chemistry, University of Adelaide, South Australia
(Received 30 March 1965) AbstmctmModels are described for the discussion of the time taken for the encounter of diffusing particles. The extent of geminate recombination of free radicals produced by the decomposition of initiator is estimated and is 10 per cent or less. IN the development of the theory of diffusion-controlled reactions it is instructive to treat reaction as the result of the diffusive meanderings of the particles. This model differs from that adopted by Smoluehowski and succeeding workers, (2) but is the basis of treatments given by Rabinovitch °) and more recently by ourselves. (1) The treatment of Rabinovitch is based upon a Lattice theory of liquids, and is more particularly limited to small particles. Our paper gave a treatment for large particles and attempted to avoid the restriction of a Lattice model. Both these treatments relate the rate of encounter to the frequency and average magnitude of diffusion jumps, which are not directly measured. It would be more appropriate to relate this rate to the directly measured diffusivity.
The probability of encounter between particles As a first step we consider the diffusion together of two particles. The probability that, in an element of time dt, the centre of mass of a particle will be found in an element of volume dn is Pdv. dt, and P is given by the general diffusion equation: D1V2pI -
dP1 dt
Such an equation holds good for each particle, designated 1 and 2. The differential equations need to be solved for appropriate initial condition by which, at a given time, the positions of the particles are specified. The equations are easily solved for the conveniently realistic case where the particles are allowed to diffuse in an infinite volume. In the case of particles that are sufficiently small to allow them to be regarded as points the state of encounter is easily defined and studied, since it is to be identified with the instantaneous coincidence of the points. A measure of the probability that this requirement of coalescence will be fulfilled at time t is Q(t) = S PiP2dv. The integration is 247
248
C.R. PATRICK and P. E. M. ALLEN
carried out over the whole volume. The integrand represents the probability that at the time of interest both particles are within the element of volume dr. Particles that are large, such as macromolecules, cannot be treated as points. Indeed such an approximate treatment is probably inappropriate even for small molecules. A different approach to the problem is therefore needed. As a first step to the solution of the more important problem it is convenient to introduce a device to simplify subsequent analysis. The motion of two moving particles of interest may be considered in the more relevant of these two particles with respect to one another. The two equations, each describing the diffusive motions of the particles with respect to an independent origin, can be replaced by a single equation set up with respect to one of the particles as origin. The new equation is of the form D1,2V2P2 = dP2/dt where P2dv is the probability of finding P2 in an element of volume dr, defined with respect to particle 1 as origin, and where D1, 2 is the modified diffusivity, equalling the sum of the diffusivities of the two particles with respect to a fixed origin. A more precise approach to the problem of particles of finite size requires the adoption of a criterion for reaction. In the case of small free radicals this is taken to be the approach of the two particles so that the centre of mass of one particle lies on the sphere, of radius equal to the collision diameter, centred on the centre of mass of the other particle. In other words, contact or encounter, rather than coalescence or coincidence is the reaction criterion. This is a realistic model since this condition implies the diffusion of the particles onto adjacent sites in the solution and, in the type of reaction we are considering, the probability o f reaction o f radicals on adjacent sites is many times greater than the probability of diffusion apart. (4) The problem ofmacroradicals is more difficult since the motion of the reactive segment of the radical must be considered as well as its centre of mass. is) However, we have shown (1) that it is a permissible approximation to treat a diffusion controlled macroradical reaction as the diffusion together of the reactive segments of the radicals. The criterion for reaction is therefore the diffusion of the reactive segments of the radicals into contact. The solution of the diffusion equation is difficult, even if the volume in which diffusion may take place is regarded as infinite, since account must be taken of the impossibility of passage through the hypothetical sphere described by the critical spherical shell. Thus, if one were to take the solution to the diffusion equation appropriate to diffusion not limited in this way, and to use this to calculate the probability of the particle lying in the defined critical relative dispositions, there is the possibility of over-estimating the correct probability, since some of the particles may be regarded as having arrived at a given point in the spherical shell by taking a path through the "excluded" or "prohibited" volume. The probability of encounter of a selected pair of particles in a real system is determined not only by the probability that their effective relative motions are correct, but also by the probability that one or other of the particles are not first removed by encounter with a third particle approaching from behind. This is not a complication in the calculations that follow since we are in the first case only concerned with the two geminate particles which can be assumed to be much nearer to each other than to any other, and in the second and more general cases the required average behaviour of all other particles may be treated as a probability density within
Diffusion-controlledReactionsin Free RadicalPolymerization
249
volume elements. Under pseudo-stationary state conditions, the density within each element may be regarded as constant although individual particles may be destroyed or created, and can be related directly to the observed concentration. Such a model as that just described differs from that of Smoluchowski et al. in that only two particles are considered. The extension of the model to realistic systems involves the averaging of the time of coalescence for particles having initially differing distance of separation, these distances being defined, in accordance with the requirement that the distribution of significant particles is uniform (or nearly so if account should be taken of initial proximity of geminate partners). This avoids the difficulties of the assumption of concentration gradients of particles towards a chosen representative, as assumed by Smoluchowski. An alternative approach is as follows. Consider one of the particles as fixed. Let the other particle be at a defined point at zero time. This particle will be free to move in any direction. The average probability that at any subsequent time it will have moved to a point within a solid angle doJ subtended at its original positions will be ~r" Let the fixed particle, or more appropriately the collision cross-section for the pair of particles centred on the centre of mass of the fixed particle, subtend at the original position of the moving particle a solid angle o~. The probability that the two particles will collide (in any relatively short finite time) is proportional to the probability that the moving particle will be found with the volume defined by the solid angle, and will be proportional to oJ/4rr. The time taken for encounter remains to be obtained from a complete solution of the diffusion equation. In extending this model to a real system containing many particles, we find that since we are interested only in first encounters, particles remote from the chosen moving particle will be obscured by closer-lying particles. Encounter is more likely to occur with the more closely lying particles. In considering more fully the situation, it is impossible to ignore relative motions of the atmosphere of particles. This, however, only complicates the estimate of probability of encounter of a particularly chosen pair of particles, one of them lying remotely: it is unlikely greatly to complicate average times of encounter.
The probability of geminate recombination in an infinite system In considering geminate recombination of radicals it is necessary to remember that this includes not only the recombination of radicals which have not escaped from the immediate molecular vicinity of the site of their creation, and consequently which cannot have initiated chains but also the recombination of chain carriers arising from a pair of radicals created together. The former may be called caged geminate recombination, the latter secondary geminate chain recombination. A further difficultyin interpretation of the overall kinetics lies in the impossibility of distinguishing between geminate recombination of the radicals within the cage and the recombination of those which escape from the cage but react with themselves or other radicals without initiating chains, which may be called secondary geminate recombination of the primary radicals. In estimating the extent of geminate recombination we have to estimate the critical solid angle subtended by the collision cross-section of one radical at the centre of mass of the other. An assurnption must be made about their separation at the beginning of their random diffusion through the system. We need to count only those instances where the
250
C.R. PATRICK and P. E. M. ALLEN
radicals escape from the primary cage. A full account of the behaviour in, and the escape from, the primary cage involves factors other than diffusion. The probability of recombination of radicals within the primary cage is many times greater than the probability of one ofthem escaping from the cage. If the estimates of Noyes~4~are taken, for instance, the initiating efficiency of, say, azo or peroxy initiators would be far lower than 60 per cent, or more, commonly found. C6~ The reason must involve either an obstacle to the reaction of the nascent radicals, due perhaps to their being in a vibrationally excited state, or, possibily, since a proportion of the activated molecules decomposing will contain more than the energy necessary to break the critical bond, some radicals will be produced with excess momenta directed away from the site of creation which may be sufficient to carry them out of the primary cage. This justifies the specification that we count only secondary geminate recombination and assume the starting point for age minate pair to be when they are separated by the diameter of one molecule. In terms of a lattice model this requires that one radical lies in the second co-ordination shell of the one chosen as origin. Assumptions must be made about the number of molecules in the second co-ordination shell and also about the number of sites which correspond to reaction. If close packing is assumed and the critical angle is assumed to be that subtended by the collision cross-section at the origin, the probability of germinate reaction (tg) is 1/11. If a looser packing is assumed, where the molecules are assumed to be arranged in shells with their centres of mass situated at integral multiples of the molecular radii from the origin, tg is 1/16. However the collision cross-section is probably not a fully appropriate criterion for rapid liquid phase reactions. It is probable that, since the rate of interchange of molecules from site to site is so much slower than their rates of oscillation about the sites, reaction is likely to occur on sites not encompassed by the collision cross-section. Consequently the critical solid angles will probably be greater and ts possibly as high as 1/5. These estimates assume that both radicals and solvent molecules are arrayed on sites of equal volume. If the radicals were larger than the molecules, estimates would have to be made on the extent to which the greater collision cross-section compensated for the looser packing. The probability of geminate recombination relative to normal termination in a finite system The foregoing estimates apply to an infinite system and give the probability of geminate relative to normal termination in such a system. In order to calculate the probability of the reference radicals being terminated within a certain distance of the origin it is necessary to calculate the probability of its termination with every radical within this distance. In order to estimate this we assume that molecules and radicals are arranged in shells, one molecule thick, centred on the origin. The probable number of radicals within each shell is given by the volume of the shell times the concentration. For the ntla shell this is: Pn = 4~rNd3n2 if n is large, d is the molecular diameter in cm, N the number of radicals per cm 3. The probability of encounter between the reference radical and another radical is represented by the critical solid angle subtended by the collision cross-section of this other radical at the origin, provided termination has not occurred in an earlier shell. Since in the system of interest, encounter will lead to termination, the probability of
Diffusion-controlled Reactions in Free Radical Polymerization
251
encounter is represented by that portion of the critical solid angle which is uneelipsed by the critical solid angles of the radicals in shells lying closer to the origin. The collision cross-section is ~rd2 and so the critical angle of one radical in the n ua shell is ~r/n 2 and the probable total critical angle of all radicals in the n th shell: ¢,, = p . lr/n 2 = 4~r2Nd 3 = ¢, n # 2
being independent of n. The second shell containing, by definition, the twin, has a probable total critical angle: ¢2
----"
¢+¢g
=
¢+~r/4
The uneclipsed portion of the probable total critical angle of all radicals in the n t~ shell is: ¢,,'
=
(1
-
e./4~r) ¢.
where the total angle of the n th shell eclipsed is E. = ¢1'+¢2'+¢3'...¢;,-,
Thus ¢.' = 4zr2 N d 3 ( 1 5 / 16 - 7rNd 3) (1 - ~rNd3) " - 2 and E . = 4 7 r [ 1 - ( 1 5 / 1 6 - r r N d 3 ) ( 1 - ~rNd3)"-2]
The probability of reaction with a radical in the n th shell is t n = ¢.'47r = r t N d 3 ( 1 5 / 1 6 - r r N d 3 ) ( 1 - ~ r N d 3 )
"-2
the probability of reaction prior to the n th shell is n--I
Y, tj = E./4rr = 1 - ( 1 5 / 1 6 - r r N d 3 ) ( l - T r N d 3 )
"-2
and the probability of reaction within x cm of the origin is tx = 1-(15/16-~rNd3)(l-TrNd3)x/d-2;
x = nd
~--- (1 + 15~rNdZx)/16, for x >> 2d and plausible values of N and d. The probability of geminate termination on this model is t~ = 1/16,
so the probability of bimolecular termination, within a distance x of the origin being geminate, is tg/t~ = 1/(1 + 15rrNd 2 x )
This is a discontinuous function, as tx cannot exceed unity. It attains a minimum value of 1/16 as N and x are increased. If a cell diameter of 2 × 10-7 cm is assumed, and a radical concentration of 1013 molecules/cm 3 (1.7× 10-sM), the minimum value is attained in 0-8 cm. DISCUSSION The values for the extent of geminate recombination, for reasons cited, can only give a rough estimate of the order of magnitude of the phenomemon. In normal-sized reaction vessels, except at very low concentrations, a calculation assuming an infinite
252
C.R. PATRICK and P. E. M. ALLEN
system is just as adequate as one assuming a finite system. This treatment however does not predict any dependence of geminate recombination on radical concentration. The deviations from normal kinetics in vinyl acetate polymerizations at low rates o f initiation observed by Bengough ~7~have been attributed to geminate chain recombination. t8~ The calculations based on a system of finite volume indicate that the concentration at which the proportion of bimolecular termination which is geminate becomes concentration dependent is of the same order as that where these deviations begin to appear. However the concentration effect indicated by the calculations may arise from a wall effect. The consequences ofwall termination in liquid-phase reactions have never been studied, but in all probability it would be first order in radical concentration. The deviation from normal kinetics observed at very low radical concentrations may well be due to the direct effect of increasing wall termination as well as to an increase in the geminate proportions of bimolecular termination. Dependence of geminate termination on concentration would be introduced if the model were modified to allow for the prior removal of the geminate radical by the prior attack of another radical from behind, an eventuality which was tacitly excluded from the calculations. The fact that geminate termination is considerable (i.e. 6--20 per cent of radical pairs formed) while deviations from the normal kinetic equation for radical polymerization are negligible implies that most of the secondary geminate termination (i.e. of radicals that have escaped from the primary cage) is in fact between the primary radicals themselves. Geminate chain combination~S) is not important under normal conditions. The only kinetic consequence is therefore a low initiator efficiency. It has been shown Ca) that at high conversion geminate chain recombination becomes increasingly important. This appears to be the result of heterogeneity (or microheterogeneity) of the highly viscous gel.tg) The present calculations apply specifically to homogeneous isotropic systems and throw no light on this problem.
REFERENCES (1) (2) (3) (4) (5) (6) (7) (8) (9)
Part II: Makromolek. Chem. 72, 106 (1964). M. yon Smoluchowski,Z. phy. Chem. 92, 129 (1917). E. Rabinovitch and Wood, Trans. Faraday Soc. 32, 547 (1936). R. M. Noyes, J. Am. chem. Soc. 77, 2042 (1955). S.W. Benson and A. M. North, J. Am. chem. Soc. 81, 1339(1959), 84, 935 (1962); A. M. North and G. A. Reed, Trans. Faraday Soc. 57, 871 (1961), J. Polym. Sci. AI, 1311 (1965). J. C. Bevington, Radical Polymerisation, Chap. II. Academic Press (1961). W. I. Bengough, Trans. Faraday Soc. 58, 1716 (1962). P. E. M. Allen and C. R. Patrick, Trans. Faraday Soc. 59, 1819 (1963). G. M. Burnett and L. D. Loan, Coll. Czech. Chem. Comm. 22, 113 (1957).
Rrsumr--L'rtude du temps necessaire/~ la rencontre des particules diffusantes est discutre sous forme de schrmas. On estime au maximum h 10~o le taux d'accouplement des radicaux libres provenant de la drcomposition du produit initial. Sommaria---Si descrivono modeUi per la discussione del tempo necessario per l'incontro di particelle in diffusione. L'intensith del ricombinamento abbinato delle radicali libere prodotto dalla decomposizione dell'iniziatore/~ valutata a ed equivale al 10~ o meno.
Zusammenfassung--Die fiir das Zusammentreffen diffundierender Partikel erforderliche Zeit wird an Hand yon Modellen besprochen. Das Ausmag der gepaarten Wiedervereinigung der durch ]nitiatorzerfall erzeugten Partikel wird gesch~itztund betrr~gt 10~0 oder weniger.
Pergamon Press Ltd.
DIFFUSION-CONTROLLED RADICAL
Printed in England.
REACTIONS
POLYMERIZATION--Ill.
IN FREE
AN ESTIMATE OF THE
IMPORTANCE OF GEMINATE RECOMBINATION OF RADICALS(1) C. R. PATmCI¢ and P. E. M. ALLEN Department of Chemistry, University, Birmingham, 15, England Department of Physical Chemistry, University of Adelaide, South Australia
(Received 30 March 1965) AbstmctmModels are described for the discussion of the time taken for the encounter of diffusing particles. The extent of geminate recombination of free radicals produced by the decomposition of initiator is estimated and is 10 per cent or less. IN the development of the theory of diffusion-controlled reactions it is instructive to treat reaction as the result of the diffusive meanderings of the particles. This model differs from that adopted by Smoluehowski and succeeding workers, (2) but is the basis of treatments given by Rabinovitch °) and more recently by ourselves. (1) The treatment of Rabinovitch is based upon a Lattice theory of liquids, and is more particularly limited to small particles. Our paper gave a treatment for large particles and attempted to avoid the restriction of a Lattice model. Both these treatments relate the rate of encounter to the frequency and average magnitude of diffusion jumps, which are not directly measured. It would be more appropriate to relate this rate to the directly measured diffusivity.
The probability of encounter between particles As a first step we consider the diffusion together of two particles. The probability that, in an element of time dt, the centre of mass of a particle will be found in an element of volume dn is Pdv. dt, and P is given by the general diffusion equation: D1V2pI -
dP1 dt
Such an equation holds good for each particle, designated 1 and 2. The differential equations need to be solved for appropriate initial condition by which, at a given time, the positions of the particles are specified. The equations are easily solved for the conveniently realistic case where the particles are allowed to diffuse in an infinite volume. In the case of particles that are sufficiently small to allow them to be regarded as points the state of encounter is easily defined and studied, since it is to be identified with the instantaneous coincidence of the points. A measure of the probability that this requirement of coalescence will be fulfilled at time t is Q(t) = S PiP2dv. The integration is 247
248
C.R. PATRICK and P. E. M. ALLEN
carried out over the whole volume. The integrand represents the probability that at the time of interest both particles are within the element of volume dr. Particles that are large, such as macromolecules, cannot be treated as points. Indeed such an approximate treatment is probably inappropriate even for small molecules. A different approach to the problem is therefore needed. As a first step to the solution of the more important problem it is convenient to introduce a device to simplify subsequent analysis. The motion of two moving particles of interest may be considered in the more relevant of these two particles with respect to one another. The two equations, each describing the diffusive motions of the particles with respect to an independent origin, can be replaced by a single equation set up with respect to one of the particles as origin. The new equation is of the form D1,2V2P2 = dP2/dt where P2dv is the probability of finding P2 in an element of volume dr, defined with respect to particle 1 as origin, and where D1, 2 is the modified diffusivity, equalling the sum of the diffusivities of the two particles with respect to a fixed origin. A more precise approach to the problem of particles of finite size requires the adoption of a criterion for reaction. In the case of small free radicals this is taken to be the approach of the two particles so that the centre of mass of one particle lies on the sphere, of radius equal to the collision diameter, centred on the centre of mass of the other particle. In other words, contact or encounter, rather than coalescence or coincidence is the reaction criterion. This is a realistic model since this condition implies the diffusion of the particles onto adjacent sites in the solution and, in the type of reaction we are considering, the probability o f reaction o f radicals on adjacent sites is many times greater than the probability of diffusion apart. (4) The problem ofmacroradicals is more difficult since the motion of the reactive segment of the radical must be considered as well as its centre of mass. is) However, we have shown (1) that it is a permissible approximation to treat a diffusion controlled macroradical reaction as the diffusion together of the reactive segments of the radicals. The criterion for reaction is therefore the diffusion of the reactive segments of the radicals into contact. The solution of the diffusion equation is difficult, even if the volume in which diffusion may take place is regarded as infinite, since account must be taken of the impossibility of passage through the hypothetical sphere described by the critical spherical shell. Thus, if one were to take the solution to the diffusion equation appropriate to diffusion not limited in this way, and to use this to calculate the probability of the particle lying in the defined critical relative dispositions, there is the possibility of over-estimating the correct probability, since some of the particles may be regarded as having arrived at a given point in the spherical shell by taking a path through the "excluded" or "prohibited" volume. The probability of encounter of a selected pair of particles in a real system is determined not only by the probability that their effective relative motions are correct, but also by the probability that one or other of the particles are not first removed by encounter with a third particle approaching from behind. This is not a complication in the calculations that follow since we are in the first case only concerned with the two geminate particles which can be assumed to be much nearer to each other than to any other, and in the second and more general cases the required average behaviour of all other particles may be treated as a probability density within
Diffusion-controlledReactionsin Free RadicalPolymerization
249
volume elements. Under pseudo-stationary state conditions, the density within each element may be regarded as constant although individual particles may be destroyed or created, and can be related directly to the observed concentration. Such a model as that just described differs from that of Smoluchowski et al. in that only two particles are considered. The extension of the model to realistic systems involves the averaging of the time of coalescence for particles having initially differing distance of separation, these distances being defined, in accordance with the requirement that the distribution of significant particles is uniform (or nearly so if account should be taken of initial proximity of geminate partners). This avoids the difficulties of the assumption of concentration gradients of particles towards a chosen representative, as assumed by Smoluchowski. An alternative approach is as follows. Consider one of the particles as fixed. Let the other particle be at a defined point at zero time. This particle will be free to move in any direction. The average probability that at any subsequent time it will have moved to a point within a solid angle doJ subtended at its original positions will be ~r" Let the fixed particle, or more appropriately the collision cross-section for the pair of particles centred on the centre of mass of the fixed particle, subtend at the original position of the moving particle a solid angle o~. The probability that the two particles will collide (in any relatively short finite time) is proportional to the probability that the moving particle will be found with the volume defined by the solid angle, and will be proportional to oJ/4rr. The time taken for encounter remains to be obtained from a complete solution of the diffusion equation. In extending this model to a real system containing many particles, we find that since we are interested only in first encounters, particles remote from the chosen moving particle will be obscured by closer-lying particles. Encounter is more likely to occur with the more closely lying particles. In considering more fully the situation, it is impossible to ignore relative motions of the atmosphere of particles. This, however, only complicates the estimate of probability of encounter of a particularly chosen pair of particles, one of them lying remotely: it is unlikely greatly to complicate average times of encounter.
The probability of geminate recombination in an infinite system In considering geminate recombination of radicals it is necessary to remember that this includes not only the recombination of radicals which have not escaped from the immediate molecular vicinity of the site of their creation, and consequently which cannot have initiated chains but also the recombination of chain carriers arising from a pair of radicals created together. The former may be called caged geminate recombination, the latter secondary geminate chain recombination. A further difficultyin interpretation of the overall kinetics lies in the impossibility of distinguishing between geminate recombination of the radicals within the cage and the recombination of those which escape from the cage but react with themselves or other radicals without initiating chains, which may be called secondary geminate recombination of the primary radicals. In estimating the extent of geminate recombination we have to estimate the critical solid angle subtended by the collision cross-section of one radical at the centre of mass of the other. An assurnption must be made about their separation at the beginning of their random diffusion through the system. We need to count only those instances where the
250
C.R. PATRICK and P. E. M. ALLEN
radicals escape from the primary cage. A full account of the behaviour in, and the escape from, the primary cage involves factors other than diffusion. The probability of recombination of radicals within the primary cage is many times greater than the probability of one ofthem escaping from the cage. If the estimates of Noyes~4~are taken, for instance, the initiating efficiency of, say, azo or peroxy initiators would be far lower than 60 per cent, or more, commonly found. C6~ The reason must involve either an obstacle to the reaction of the nascent radicals, due perhaps to their being in a vibrationally excited state, or, possibily, since a proportion of the activated molecules decomposing will contain more than the energy necessary to break the critical bond, some radicals will be produced with excess momenta directed away from the site of creation which may be sufficient to carry them out of the primary cage. This justifies the specification that we count only secondary geminate recombination and assume the starting point for age minate pair to be when they are separated by the diameter of one molecule. In terms of a lattice model this requires that one radical lies in the second co-ordination shell of the one chosen as origin. Assumptions must be made about the number of molecules in the second co-ordination shell and also about the number of sites which correspond to reaction. If close packing is assumed and the critical angle is assumed to be that subtended by the collision cross-section at the origin, the probability of germinate reaction (tg) is 1/11. If a looser packing is assumed, where the molecules are assumed to be arranged in shells with their centres of mass situated at integral multiples of the molecular radii from the origin, tg is 1/16. However the collision cross-section is probably not a fully appropriate criterion for rapid liquid phase reactions. It is probable that, since the rate of interchange of molecules from site to site is so much slower than their rates of oscillation about the sites, reaction is likely to occur on sites not encompassed by the collision cross-section. Consequently the critical solid angles will probably be greater and ts possibly as high as 1/5. These estimates assume that both radicals and solvent molecules are arrayed on sites of equal volume. If the radicals were larger than the molecules, estimates would have to be made on the extent to which the greater collision cross-section compensated for the looser packing. The probability of geminate recombination relative to normal termination in a finite system The foregoing estimates apply to an infinite system and give the probability of geminate relative to normal termination in such a system. In order to calculate the probability of the reference radicals being terminated within a certain distance of the origin it is necessary to calculate the probability of its termination with every radical within this distance. In order to estimate this we assume that molecules and radicals are arranged in shells, one molecule thick, centred on the origin. The probable number of radicals within each shell is given by the volume of the shell times the concentration. For the ntla shell this is: Pn = 4~rNd3n2 if n is large, d is the molecular diameter in cm, N the number of radicals per cm 3. The probability of encounter between the reference radical and another radical is represented by the critical solid angle subtended by the collision cross-section of this other radical at the origin, provided termination has not occurred in an earlier shell. Since in the system of interest, encounter will lead to termination, the probability of
Diffusion-controlled Reactions in Free Radical Polymerization
251
encounter is represented by that portion of the critical solid angle which is uneelipsed by the critical solid angles of the radicals in shells lying closer to the origin. The collision cross-section is ~rd2 and so the critical angle of one radical in the n ua shell is ~r/n 2 and the probable total critical angle of all radicals in the n th shell: ¢,, = p . lr/n 2 = 4~r2Nd 3 = ¢, n # 2
being independent of n. The second shell containing, by definition, the twin, has a probable total critical angle: ¢2
----"
¢+¢g
=
¢+~r/4
The uneclipsed portion of the probable total critical angle of all radicals in the n t~ shell is: ¢,,'
=
(1
-
e./4~r) ¢.
where the total angle of the n th shell eclipsed is E. = ¢1'+¢2'+¢3'...¢;,-,
Thus ¢.' = 4zr2 N d 3 ( 1 5 / 16 - 7rNd 3) (1 - ~rNd3) " - 2 and E . = 4 7 r [ 1 - ( 1 5 / 1 6 - r r N d 3 ) ( 1 - ~rNd3)"-2]
The probability of reaction with a radical in the n th shell is t n = ¢.'47r = r t N d 3 ( 1 5 / 1 6 - r r N d 3 ) ( 1 - ~ r N d 3 )
"-2
the probability of reaction prior to the n th shell is n--I
Y, tj = E./4rr = 1 - ( 1 5 / 1 6 - r r N d 3 ) ( l - T r N d 3 )
"-2
and the probability of reaction within x cm of the origin is tx = 1-(15/16-~rNd3)(l-TrNd3)x/d-2;
x = nd
~--- (1 + 15~rNdZx)/16, for x >> 2d and plausible values of N and d. The probability of geminate termination on this model is t~ = 1/16,
so the probability of bimolecular termination, within a distance x of the origin being geminate, is tg/t~ = 1/(1 + 15rrNd 2 x )
This is a discontinuous function, as tx cannot exceed unity. It attains a minimum value of 1/16 as N and x are increased. If a cell diameter of 2 × 10-7 cm is assumed, and a radical concentration of 1013 molecules/cm 3 (1.7× 10-sM), the minimum value is attained in 0-8 cm. DISCUSSION The values for the extent of geminate recombination, for reasons cited, can only give a rough estimate of the order of magnitude of the phenomemon. In normal-sized reaction vessels, except at very low concentrations, a calculation assuming an infinite
252
C.R. PATRICK and P. E. M. ALLEN
system is just as adequate as one assuming a finite system. This treatment however does not predict any dependence of geminate recombination on radical concentration. The deviations from normal kinetics in vinyl acetate polymerizations at low rates o f initiation observed by Bengough ~7~have been attributed to geminate chain recombination. t8~ The calculations based on a system of finite volume indicate that the concentration at which the proportion of bimolecular termination which is geminate becomes concentration dependent is of the same order as that where these deviations begin to appear. However the concentration effect indicated by the calculations may arise from a wall effect. The consequences ofwall termination in liquid-phase reactions have never been studied, but in all probability it would be first order in radical concentration. The deviation from normal kinetics observed at very low radical concentrations may well be due to the direct effect of increasing wall termination as well as to an increase in the geminate proportions of bimolecular termination. Dependence of geminate termination on concentration would be introduced if the model were modified to allow for the prior removal of the geminate radical by the prior attack of another radical from behind, an eventuality which was tacitly excluded from the calculations. The fact that geminate termination is considerable (i.e. 6--20 per cent of radical pairs formed) while deviations from the normal kinetic equation for radical polymerization are negligible implies that most of the secondary geminate termination (i.e. of radicals that have escaped from the primary cage) is in fact between the primary radicals themselves. Geminate chain combination~S) is not important under normal conditions. The only kinetic consequence is therefore a low initiator efficiency. It has been shown Ca) that at high conversion geminate chain recombination becomes increasingly important. This appears to be the result of heterogeneity (or microheterogeneity) of the highly viscous gel.tg) The present calculations apply specifically to homogeneous isotropic systems and throw no light on this problem.
REFERENCES (1) (2) (3) (4) (5) (6) (7) (8) (9)
Part II: Makromolek. Chem. 72, 106 (1964). M. yon Smoluchowski,Z. phy. Chem. 92, 129 (1917). E. Rabinovitch and Wood, Trans. Faraday Soc. 32, 547 (1936). R. M. Noyes, J. Am. chem. Soc. 77, 2042 (1955). S.W. Benson and A. M. North, J. Am. chem. Soc. 81, 1339(1959), 84, 935 (1962); A. M. North and G. A. Reed, Trans. Faraday Soc. 57, 871 (1961), J. Polym. Sci. AI, 1311 (1965). J. C. Bevington, Radical Polymerisation, Chap. II. Academic Press (1961). W. I. Bengough, Trans. Faraday Soc. 58, 1716 (1962). P. E. M. Allen and C. R. Patrick, Trans. Faraday Soc. 59, 1819 (1963). G. M. Burnett and L. D. Loan, Coll. Czech. Chem. Comm. 22, 113 (1957).
Rrsumr--L'rtude du temps necessaire/~ la rencontre des particules diffusantes est discutre sous forme de schrmas. On estime au maximum h 10~o le taux d'accouplement des radicaux libres provenant de la drcomposition du produit initial. Sommaria---Si descrivono modeUi per la discussione del tempo necessario per l'incontro di particelle in diffusione. L'intensith del ricombinamento abbinato delle radicali libere prodotto dalla decomposizione dell'iniziatore/~ valutata a ed equivale al 10~ o meno.
Zusammenfassung--Die fiir das Zusammentreffen diffundierender Partikel erforderliche Zeit wird an Hand yon Modellen besprochen. Das Ausmag der gepaarten Wiedervereinigung der durch ]nitiatorzerfall erzeugten Partikel wird gesch~itztund betrr~gt 10~0 oder weniger.