Makromol. Chem. 183,2191 -2211 (1982) 2191

Determination of Network Structure by Extraction and Random Degradation, l a )

Theory

Dedicated to Professor Dr. Dres. h.c. G. V. Schulz on his 75th birthday

Martin Hoffmann

Corporate Research Division of Bayer AG, D-5090 Leverkusen, Federal Republic of Germany

(Date of receipt: June 15, 1981)

SUMMARY: The weight fraction and the molecular massb) of polymeric material extracted from a polymer

network, the time needed for the dissolution of the network during random degradation, and the molecular mass of the high molecular mass degradation products can be utilised for the determination of previously inaccessible structural details of the network, e. g. the value of the molecular mass of network chains, the chain end correction term, the functionality of the cross- links, and the inhomogeneity of crosslinking, i. e. the amount, the diameter, and the crosslink concentration of anomalously and highly crosslinked regions.

1. Introduction

The analytical methods currently used to determine parameters of the network structure do not, as a rule, permit any definite statements to be made as to the con- centration of crosslinking units, any local inhomogeneity, the functionality of the crosslinks, or the number of free chain ends in the network.

Chemical or spectroscopic determination of the concentration of crosslinking units is generally not sensitive enough to give reliable results for concentrations of about lo-’ w t . 4 , particularly since the crosslinks often are chemically not much different from the normal chain elements, and because all the occurring types of crosslinks are not always known. In addition to this, such methods of detection often measure also those molecules of a bifunctional reagent which are chemically bonded to a chain with only one function, or which are not bonded at all. Finally, the possibility that the detection reaction may bring about unwanted changes in the structure which is to be analyzed cannot always be ruled out. Local inhomogeneities can generally only be determined by such methods if they have sizes z0,l mm.

Quantitative conclusions relating to the parameters of the network structure can only be drawn in exceptional cases from the kinetics of the crosslinking reaction; this is because simplifying assumptions about the mechanism often have to be introduced, in order to be able to carry out kinetic calculations, and because important side-reactions are not always sufficient- ly well-known, particularly in the case of technical crosslinking processes.

Even the widely used methods of elasticity and swelling measurements do not permit any sufficiently reliable conclusions to be drawn as to the concentration of crosslinking units, because the relationships between the structure and the constants of elastic behavior depend on

a) Parts 2 and 3: cf.’,*). b, “Relative molecular mass” (systematic IUPAC name) is shortened to “molecular mass”

throughout this paper.

0025-1 16)3/82/09 2191-21/$03.00

2192 M. Hoffmann

the method of measurement (e. g. strain rate) in a way which has yet to be adequately explained. Furthermore the methods of evaluation have not, as yet, been theoretically substantiated with any adequate degree of certainty. Thus many aspects of a determination of the network structure by such measurements are still unclear2).

In view of the situation, it appeared necessary to develop new methods for deter- mining network structures. The random process of degradation of network chains and the investigationn of molecular mass distribution and branching of the high- molecular-mass soluble scission products of the networks, as well as characterization of extractable components, proved to be particularly suitable methods.

2. Structural Parameters of Networks and Definitions

Networks are branched macromolecules which contain cyclic chains. If particles structured in this way have masses <lob6 grams, they are usually designated as microgel. Structurally ideal networks can be defined by the following conditions: All the network chains have the same molecular mass M,, the same numberf, of network chains originates from each crosslink, there are no chain ends, all the network chains have the same statistical end-to-end distance ( g ) l l 2 as free molecules of the same molecular mass under otherwise same thermodynamical conditions. Simple networks are those in which each network chain leads to its spatially nearest crosslink and not to a more distant one. The diamond lattice and the cube lattice represent ideal and simple networks of this sort, if the atomic bonds are replaced by h,-vectors and some thermal movement is allowed.

Real networks are more irregular and have a more complex structure. Fig. 1 demonstrates this by showing a part of a two-dimensional network which is still simple (see below). Here, the chains themselves are not drawn, but only their vectors hi, whose lengths are subject to fluctuations with time and whose momentary lengths in Fig. 1 are not proportional to the molecular mass of the network chain The values of Mc,i exhibit a distribution which depends on the type of the crosslinking process and should, in many cases, be a Schulz-Flory d i s t r ibu t i~n~*~) with the weight fractions wi (M,,i) :

The weight fractions w , , ~ depend on Mc,i and on the number average The corresponding weight average is Mc,w = 2M,,,. In 1 cm3 of the pure, crosslinked polymer which contains no free chain ends, but possibly some elastically ineffective loops, there are N, = pNA/MC,. network chains, p being the density of the polymer and NA Avogadro’s number.

The functionality f, is also determined by the crosslinking process and may (for reasons which will be discussed in Part 2l)) be abnormally high at certain points, e. g. at the encircled crosslinks shown in Fig. I . Furthermore, the functionality may assume a value which is abnormally high with respect to the elastic effect if (see

Determination of Network Structure by Extraction and Random Degradation, 1 21 93

Fig. 1 . A nonideal two-dimensional network which is still considered to be simple (see text)

rectangle in Fig. 1) two crosslinks are connected by a very short chain (Mc MCJ, while the normal network chains are very much longer. This may occur when free radicals polymerize double bonds of more than one chain. With real networks, there- fore, a distribution of functionalities is to be expected, which may be due to a certain type of inhomogeneous distribution of the crosslinks. The average functionality, x, and Mc,n determine the number of crosslinks per cm3, namely 2Nc/ f ; , provided each chain joins two different crosslinks.

Furthermore volume elements with an abnormally high concentration of crosslinks (see e. g. the centre of Fig. 1) may have diameters considerably higher than h, so that, with regard to the elastic effect, they can no longer be treated as crosslinks of higher functionality. The specimen is then inhomogeneously crosslinked and has a non- uniform spatial distribution of the concentration of crosslinking units. It is sometimes sufficient to distinguish between two volume fractions differing in the concentrations of crosslinking units and to consider the larger one as the matrix in which the smaller one is dispersed. In discussing the properties of inhomogeneously crosslinked specimens of this type the spatial distribution of the dispersed fraction must also be taken into account.

Most crosslinking processes may lead to the formation of loops. A loop is part of a chain, which starts from one crosslink and returns to it without being connected to other crosslinks.

In crosslinking with peroxides, each peroxide molecule decomposes into two radicals, both of them starting their attack on the polymer in the same small volume. So it may well occur that both attack parts of the same chain and that the two polymer radicals thus formed recombine to

21 94 M. Hoffmann

build a tetrafunctional crosslink between parts of the same chain. The probability for the formation of such loops depends on the conformation of the chains, i. e. on the probability6) Pi that a statistical segment i of a chain is present in the vicinity of a certain segment of the same chain, to which we assign the number 0. At x = 0, y = 0, z = 0 we take P, = 1 and sum up the Pi over all i up to i,, = 100, i. e. a typical number of segments in network chains. Since back- coiling of segments occurs from two sides of segment 0 we have to multiply the sum with 2. Consequently about 5 of the 12 next neighbours of segment 0 are segments of the same chain’). Therefore about 5/12 of all crosslinks may be assumed to be loops, if the polymer is not diluted by a solvent. As there are 2N,/f , crosslinks connecting different chains, we should have (1017) . NJf, loops. Loops with i < 3 are so small that they are not penetrated by other chains of the melt. Therefore we may neglect them in the discussion of the properties of networks. The mean value i of all the loops with i 2 3 is calculated by Eq. (2)

100

- in has a value of about 15.

The number of loops with i > 3 is approximately (2,23/12) of all crosslinks including loops, i. e. about 0,63 * N,/f,. Their mass fraction w, may be calculated from their number by multiplying with their mass i.e. with 15Ms/NA to be 0,63p15Ms/(f,M,), which usually assumes values of about In melts about 7 other chains together with their back-coiled segments penetrate each loop of size b.

Furthermore, in a real network, chains occur of which only one chain end is bound to the network with the consequence that they are elastically ineffective. Their weight fraction we comprises both chain ends of all the molecules of the original polymer prior to crosslinking, but also the chain ends which have originated from Z, chain scissions per cm3 after crosslinking. The weight fraction winel also contains the weight fraction wsbr of the short-chain (Msbr < M,) branches of the polymer, and the weight fraction w, of the loops. If the uncrosslinked polymer has n - 2 long chain branches and two chain ends per molecule of molecular mass Mn,n (for M,,,, see Eq. (12)), we may assume that the distribution ofM,,, /M,, , crosslinks in such a molecule leads to n chains with one free end, each having a molecular mass M,,,/2. This is easily conceived in the case of linear chains which consist of two chain ends each with molecular mass M,/2 and of n - 1 unbranched parts each with molecular mass M,. The Z, scissions give rise to 2 2 , chain ends each with a molecular mass M,,,/2. Therefore we find:

Each chain with one free end is attached to a branching point which then has an elastically effective functionality fv,el equal to f , - 1 . Iff, - 1 < 3 , this branching point can no longer be considered as an elastically effective crosslink. For a small weight fraction we the elastically effective functionality has a mean value f , ( 1 - we). When f , = 4 most of the loops are not situated at elastically effective crosslinks.

Determination of Network Structure by Extraction and Random Degradation, 1 2195

Therefore Mc,n (wl + wsbr) i s the molecular mass of the elastically ineffective part of the network chains. For f, > 4 the loops are attached to crosslinks so that their elastically effective functionality is reduced tof, - 0,63.

Real networks almost always contain a small amount of short molecules which are not covalently bound to the network - i. e. an extractable component. For such in- completely crosslinked polymers, the quantity Mn,n must be determined analytically. It denotes the unknown average molecular mass of that component of the original molecular mass distribution which is not extractable after crosslinking (see Eq. (12)).

The network chains have statistical conformations whose end-to-end distances, hi, for a certain network chain molecular mass Mc,i, are distributed in such a way that their average value @ may correspond to that of free molecules (q,) of the same molecular mass in the same environment. However, the distribution of the hi-values obtained by a real crosslinking reaction may differ from the distribution of the hi,o, because fixing of the network chain ends at the crosslinks restricts the motions of each chain, and because non-ideal conditions may have prevailed during crosslinking, e. g. a slowly relaxing orientation caused by the filling of a mold. Furthermore, the polymer may have been crosslinked at a temperature higher than the temperature at which its structure is measured, or in the presence of a solvent which was subsequent- ly evaporated. There is, therefore, no reason why the so-called memory term h f / h t , should always have a value near unity. The memory term will be determined in Part 3*) at least approximately.

Moreover, in a real network the vectors hi will not usually lead to the spatially nearest crosslink, but to one of the N, . 2/fv crosslinks further apart; thus one has to consider such networks as being composed of several simple networks penetrating one another'). The structural parameter S3 = (h2)3'2. 2N, / fv characterizes the influence of the interlacing of S3 simple networks with a functionality f,. S is the ratio of h t o the mean spatial distance of crosslinks.

_ _

-

The crosslinking process leads to networks via the formation of rings. At all functionalities of the crosslinks, elastically effective rings may be formed from two or more network chains. In a network withf, = 4, rings with 3 and 4 network chains will be formed with similar probability. But in a two-dimensional network withf. = 4, rings consisting of 3 chains do not appear to be randomly distributed but are correlated with rings of 5 chains and appear in pairs: Fig. 1, upper right corner. An elastically active ring consisting of two chains is built up (left corner), if a chain leading from crosslink A to crosslink B folds back to crosslink A. In a plane network withf, = 4 this occurs with a probability smaller than 118. In a three-dimensional real network the number of neighbouring (or accessible) crosslinks is much larger than 8 so that the probability of an elastically effective backfolding of this type is small.

3. Characterisation of Networks by the Extractable Material and its Distribution of Molecular Masses

When a polymer is crosslinked to a network with network chains having the molecular mass M,, some of the molecules will not be attached to the network, especially when their molecular mass is less than M,. If their weight fraction we, is less than a few (e.g. b) per cent, then each molecule which has undergone at least one

2196 M. Hoffmann

crosslinking step will be bound almost exclusively (i. e. by (100 - b) 070) to the net- work and not to an as yet still free molecule. Looking for the molecules with M = M, we ask how many of them (number fraction po) are not bound to the network, and how many are bound by one (fractionp, ), or several crosslinks (p2, p3 . . .). For we, c 0,Ol the value of pi may be calculated by using the well-known analogy that, if one puts one hand Z times into a pot containing N balls, picks out one ball and puts it back again, then the probability pj that the same ball will be picked j times and not picked out (2 - j ) times is:

Z j ! ( Z - j)! ; P o = (1 -+) (4)

with 1 - - = l/e and in our picture Z = pNA/M, and N = pNA/M, i.e. ( i3N Z/N = M / i c one obtains for j = 0 the probability po :

P , = P , . - = P o . ( g ) Z N

2 P 2 = P q F = P 0 - ~ - ( g ) 1 z 1

3 1 z 1 P3 = P 2 . 7 ” = P o . % ’ ($)

(7)

Naturally Cpj = 1. For M = M, the fraction po = l /e of all molecules is not attached to the network. This fraction should increase if a part of the crosslinking steps leads to elastically ineffective loops and if all the molecules bearing loops are extracted. But for f , = 4 this is already taken into account by calculating M/M, with the experimental value of M,, which is increased by loops and side chains beyond the value belonging to the elastically active part of the network chain. Furthermore it is more likely that molecules bearing loops are not extracted if these loops enclose network chains and are thus attached to the network. As we have 0,63 N,/f, loops per cm3 containing N, network chains, the fraction 0,63 * Mi/cf, - M,) of the molecules with the molecular mass Mi bear at least one loop and will not be extracted.

In order to calculate the unbound and extractable amount of polymeric material of a crosslinked polymer one has to assume that the original polymer exhibits a certain type of molecular mass distribution, e.g. a Schulz distribution with the coupling constant k4):

In the abscence of loops the very small extractable fraction w , , , ~ ~ ~ may then be calculated by multiplication of wi with exp( -Mi /M, ) and integration:

Determination of Network Structure by Extraction and Random Degradation, 1 21 91

T ( k + 1): Gamma function of k + 1 . It has been tabulated9) and is equal to k! for integer values of k.

Eq. (10) is in approximate agreement with published theories"). In case of a dis- tribution of Mc values it is necessary to replace M, in Eqs. (4) - (8) and (10) - (12) by the number average molecular mass MC,,,.

The amount (in moles) nex,poly of extracted polymeric material may be calculated in a similar way; wex/n, then is the number average molecular mass of the extracted material.

For we, > 0,Ol the numerical value 1 in the denominator of Eqs. (10) and (1 1) must be substituted by 1 - wex. From M,,,,, M,, and wex,poly we calculate the number average molecular mass of the part of the polymer which formed the network.

M,,,,, = M,, ( - ) k/(l + k ) ' - "'ex,poly

The molecular mass in the maximum of the distribution (being integrated in Eq. (10)) may be found to be equal to M,,ex by calculating the first derivative. But it should be noted in a gelchromatographic determination of Mn,ex that the maximum of the elution curve does not coincide with Mn.ex (see Appendix I). Therefore we have

MGPC,ex = (1 + k)Mc,n (13)

k + 1 may be determined by Eq. (10) from the slope of the curve representing the values log we,,poly versus log M, or versus log MGPC,ex of a series of networks made from the same polymer. For many cases k = 1: Then M,/M,,,, 2: 1/= even at higher wex. If k is known, the values M,,, may be found from wex,po,y and M,. The value for M,,,, may differ appreciably from the number average molecular mass of the polymer measured before preparing the crosslinking mixture, because of chain scissions during the preparation of the crosslinking mixture. Mn,n should be used for calculating the term correcting the modulus of elasticity for elastically ineffective chain ends. But the calculated term 1 - (M,/M,,,,,) does not contain the effect of chain ends, which appeared as a consequence of long-chain branching and of chain scissions during or after crosslinkinglO).

Insofar as the amount of free chain ends is a consequence of the crosslinking process it does not change the equations derived. But the amount of extractable material will be enlarged by a fraction wl,o of degradation products if some chain scissions occurred after crosslinking. For a random splitting by 2, scissions w1 will

2198 M. Hoffmann

be calculated in section 4.2.2; it has to be added to wex,poly of Eq. (10). With po = 1 - (Zo/Nc) we find for a network with a distribution of Mc,i values and for small Zo /N , :

Eq. (14) differs from earlier theories'O) describing the influence of chain scissions on wex. Usually w , , ~ 6 we,, and the molecular masses of the degradation products are much higher than k OM,,,. Therefore they do not alter the position of the maximum of the distribution of the extracted molecules.

In order to evaluate the influence of loops we substitutep, in Eqs. (5) and (10) by p,, - [I - (0,63 M / ( f , .M,))] . From this follows that the right-hand side of Eq. (10) has to be multiplied by 1 - (2 0,63/ f , ) , and the right-hand side of Eq. (13) has to be divided by 1 + (0,63/ f,). For statistically coiled chains the corrections are small, and w e x , p o l y / ~ , e x is not changed.

4. Random Degradation of Polymer Networks

Network chains may be split at random until the network is degraded to a soluble material. The amount of soluble material as a function of degradation time, its molecular mass, and the index of branching make it possible to draw conclusions concerning the concentration of crosslinks and the inhomogeneity of crosslinking, the functionality, f, of the crosslinks, and the number and length of the elastically ineffective chain ends.

4.1 The overaN rate of degradation

4.1.1 The number of split chains with Z scissions per cm3

Before the mass fractions and molecular masses of the degradation products and the degradation time of the network can be calculated theoretically, it is necessary to explain to what extent the network chains are split once, several times, or not at all, if 2 scissions have taken place which are distributed at random in 1 cm3 of the pure crosslinked polymer and which mainly have not occurred at the crosslinks. Let us first consider the case of an ideal network with the uniform network chain molecular mass M, and N, = pN,/M, network chains per cm3, of which f, start from one crosslink. The numerical proportions (which in this case are also equal to the weight propor- tions) of the network chains which have not been split at all (= po) or once (= p,) or several times (pZ,p3.. .) can be calculated by using Eqs. (5) - (8) with N, instead of N.

Considering the case of a distribution of molecular masses Mc,i we must note that for each molecular mass Mc,i a different (Z/Nc)i must be used:

Determination of Network Structure by Extraction and Random Degradation, 1 2199

If the mass distribution of Mc,i corresponds to Eq. (I), we can calculate the fre- quency distribution of the amount (in moles) of chains of type i (i.e. n c , i = Nc,i/NA = P / M , , ~ ) in 1 cm3 from the weight fractions wc,i by dividing by Mc,i/p.

Hence it follows that the amount nunsplit of unsplit network chains in 1 cm3 of the polymer is:

m

Thus, with non-uniform M c , i and a given 2, the probability (numerical proportion) po,unsplit - nunsplit/n has a value different from that calculated with Mc,n and Eq. ( 5 ) : -

1

1 + ZINC po = ~

Similarly, one can calculate

Z/Nc - Z/Nc =Po- l + Z/Nc (1 + ZINC)’ 4 =

(ZINC)’ ZINC P z = (1 + Z/NJ3 ‘ P o ( 1 +ZINC )

Of course, once again Cpi = 1. The proportion 3 of all the chains which are already split is &/(I - Po). The weight fraction wunsplit of unsplit chains is similarly calculated:

1 (1 + Z/NC)’ Wunsplit = (20)

The number average of the network chain molecular masses of the unsplit chains is P wunsplit/nunsplit

- Mc, n

1 + ZINC Mc,unsplit,n -

For the split network chains one calculates the number average molecular mass Mc,split,n before splitting, using wsplit = w - wunsplit and nsplit = n - nunsplit :

For small ZINC, the value of Mc,split,n is equal to 2Mc,n = M,,,, but if Z/Nc 1, then it is equal to Mc,n; Mc,unsp,it,n decreases even more strongly with Z/N, than

Scissions divide a network chain into parts with mean molecular masses Mbr. If, before splitting, the Mc,i had a distribution with the non-uniformity U = 1, then the Mbr will also have a distribution with U = 1 . If Z/Nc is sufficiently high, some of the fragments of network chains produced by the first splitting (namely fraction p z / (PI + pz + . . .) will be shortened as a result of a second scission, so that the number average molecular mass Eb, of the network chain fragments decreases with decreasing Po. For uniform (index u) Mc the following applies:

Mc, split,n*

22011 M. Hoffmann

In case of non-uniform (index nu) Mc of the network, it should be noted that, in accordance with Eq. (22), the molecular mass of the split network chains is greater than Mc,n so that M, must be replaced by Mc,split,n.

P 2 +

3(1 -&) 4(1 -&)

- Mbr,nu =

In case of small Z/N, and uniform M,, one obtains 2Mb,,/M, = 1 , but 2kfbr,nu/Mc,n = 2 for non-uniform M,. In nonideal networks the quantities w, and we influence the results (see Eqs. (30) and (31)), but not the form of the equations.

4.1.2 Scissions required for dissolution of the network

At which value of the probabilityp, (see section 4.1.1) does the network break up into small fragments, in other words dissolve in a suitable solvent? We term this value PO+ and use the index + for all quantities depending on PO+. The dissolution occurs when there are no sufficiently long and branched sequences of unsplit network chains which extend from one side of the specimen to the other; po is the probability that a network chain is not split. If a sequence of unsplit chains reaches a crosslink with the functionality f,, then f, - 1 ways are possible for the continuation of the sequence. If (f, - 1) .po 2 1 then the sequence will be continued. Such sequences will be the more heavily branched, the more (f, - 1) *po exceeds unity. Long sequences of unsplit chains can no longer be formed if (f, - 1) -po < 1 . The dissolution of the network, therefore, takes place at a value of pO+ which obeys the following condition:

The greater the functionality of the crosslinks in a network, the smaller the pJ at which it will decompose (i.e. the greater the degree of degradation of the network chains). When f, varies within the network, the crosslinks with smallerf, are split off first. If they form the matrix in which individual crosslinks of higher functionality are dispersed, then the matrix and also the network will dissolve at a value of pO+ which is appropriate of the matrix.

Let uz be the rate of degradation, measured in scissions per second per cm3. Then uz may be determined from the change in M, of linear molecules:

Determination of Network Structure by Extraction and Random Degradation, 1 2201

If uz is constant during the degradation process and Zo = 0, then Z = t iz - t . Thus the time t+ required for dissolution of the network can be calculated with Z + / u z by taking Z + / N , fromp; using Eqs. ( 5 ) or (16).

for uniform M , : (%)u = -1npo+ = l n ( jv - 1) (27)

(28) 1

for non-uniform M,: 1 = j v - 2

If a network contains the weight fraction we of elastically ineffective chains with one free end, then less scissions are necessary for dissolving the network than at we = 0, because the free chain ends may be supposed to be created by scissions before the process of degradation began. Therefore the time required for dissolution is reduced by a factor 1 - we. The effect of short chain branches is usually included in the value of uz.

Furthermore we calculate the influence of loops. The fraction w, of all scissions does not lead to a scission of the elastically effective parts of the network. Therefore the number of scissions necessary for dissolving a network is larger by a factor 1/(1 - y ) than for w, = 0. But we have to consider also that loops act like polyfunc- tional crosslinks by enclosing network chains. Because of the small molecular mass of the loops they will not be split at p; . Therefore they prevent the network from being dissolved at t+ . Then the degradation has to be continued until the long (non-linear) sequences of unbroken network chains prevailing at t+ have been degraded to molecules with molecular masses around 4M,,, which may be able to slip out of the loops. This will be achieved at = t+ + At. With Eq. (26) we calculate At:

Therefore we obtain Eq. (30) for uniform M,:

For non-uniform M, the following equation holds :

The time tacPp required for the dissolution of a network depends strongly onf , and M, and weekly on the presence of loops. Thus, if v z , we and& are known, thenM,,, can be calculated from tZPp. If, on the other hand, u z , P , and M,,,/(l - we) (e.g. from extraction) are known, then we may calculate (f, - 2). With pi or (Z/N)+ we can calculate also M& . Values of 2M&/M, for pa = p; are shown in Tab. 1.

2202 M. Hoffmann

Tab. 1. Characteristic parameters of the molecules and of the size distributions of polymeric degradation products of networks with different functionalities f, at the dissolution point of the network. (For explanation of the symbols see text at the indicated equations.)

1,1589 2,3177 1,994 1,7383 0,1327 2,42 0,918 0,93 3,80 2 3

0,8639 2,5917 2,194 1,7278 0,1419 2,16 0,860 0,88 3900 2,9

0,6071 3,0354 2,504 1,8212 0,1051 2,17 0,785 0,81 2,60 2 8

0,4839 3,3873 2,738 1,9356 0,0798 2,23 0,737 0,76 2,39 2,1

0,886 0,820 0,742 -

2,66 3,28 4,46 5 3 6

4.2 Molecular mass distribution and branching index of soluble polymeric degradation products of networks

4.2.1 Molecular masses and branching indices of individual scission products

At first we shall calculate the molecular masses, branchings, and amounts of net- work fragments which are split off from an ideal network at different ZINC or po .

For uniform f v , we = 0, and w, = 0, the number average molecular masses Ml = fvMbr of the smallest branched fragments of degraded networks may be calcu- lated. For a short degradation, i. e. for po = 1 , we obtain

MC for uniform M,: 4,u = f ; - 2

for non-uniform Mc: MI,,, = f , . M,.+

At the dissolution point of the network, i. e. at p i , we find

Determination of Network Structure by Extraction and Random Degradation, 1 2203

The next larger fragments have a molecular mass M2 and contain at least one unbroken network chain. For p$ and the i-th type of fragments we derive the following number averages Mi' :

With larger f, or i a few molecules may contain more than (i - 1) unbroken net- work chains, but this may be neglected.

In order to calculate the radius of gyration rbr and the intrinsic viscosity [q]br of these branched molecules we have to calculate the weight fraction Wb, of material in the side chains which are attached to the longest sequence of unbroken network chains including two chain ends. As an approximation we then find [q]br,i from Wbr,i

and the Staudinger index [q]lin,i of linear molecules having the same molecular mass Mi as the branched ones:

a characterizes the interaction of the chains with the solvent. In thermodynamically good solvents a assumes values of about 0,72, in theta-solvents a = 0,5. The radius of gyration fbr is smaller than rlin and may be calculated with the aid of the branching parameter g b , , which is approximately equal to (1 - wbr):

We shall see in section 4.3 that each fragment i has a rather narrow distribution of molecular masses. Therefore Mi may be used to calculate [q]i,lin in Eq. (38).

4.2.2 Mass fractions of individual scission products

The weight fraction wi of a fragment i depends on the probability of its formation; (1 - po) is the probability that a chain has been split at least once, (1 - p,,po)" is then the probability that x chains selected at random are present in a split state. Sequences of x chains which surround a network fragment and release it when they are split will now be considered.

In order to obtain the weight fraction of such fragments, the probability of such a sequence must be multiplied by the number of such sequences in 1 cm3 of pure network and by the mass of one fragment. This will be demonstrated, taking as an example a network with the uniform functionalityf, = 4.

2204 M. Hoffmann

Fragments with molecular masses 4Mbr are formed as starshaped molecules with four branches. Each of the N/2 crosslinks bears 4 chains whose scission produces this fragment. Each sequence of 4 scissions is produced with the probability (1 - The weight fraction of these fragments is w,. The density of the polymer is p = NcMc/NA. With Eqs. (22) and (23) one obtains:

In a similar way one obtains the result that the formation of the next larger fragments (two interconnected stars) takes place with the probabilityp, (1 - P,)~, and corresponding sequences of bonds which are to be split occur four times at each of the Nc/2 crosslinks. If all the sequences are counted, however, each one will be counted twice, because each comprises two crosslinks, Thus there are (Nc/2) . (4/2) = Nc such sequences. The weight fraction w2 of such

split-off fragments is calculated with M2 = Mc,un,p,it,n + 6Mbr = 8Mbr 1 - M2 . We find: Mbr

Similarly, for the next larger fragment, we obtain for f , = 4:

With even larger fragments, difficulties arise, because the fragments may contain rings, and therefore different scission probabilities occur. For very long sequences of split chains, further complications arise due to the splitting off of network fragments which have already lost some material from within their interior by scission reactions.

If the degradation proceeds to a particular value of p,, therefore, the weight fraction of the soluble material obtained will be the smaller the higher the functionalityf,, provided that Z, = 0, that all the material not bound to the network has been extracted before degradation and that the fragments have been completely extracted after degradation. If Mc,n is known, then it should be possible to calculatef, from the quantity of such fragments in accordance with Eq. (42) at least for small (1 - p,) and for uniformf,. It should also be possible to use the molecular mass of these degradation products for this purpose.

Approximate values of wi of these and of larger fragments may be taken from the kinetics of condensation of polyfunctional molecules 12). The published values have to be corrected because Mi depends on the degradation time. By multiplication with 2pMi/(i. f, .M,,,) we find for w: at the dissolution point of the network:

In nonideal networks there may be crosslinks, bearing a chain with a free end. The probability that a fragment containing such a crosslink is split off is higher than that for a usual crosslink by the factor 1/(1 - p,).

Using this factor and the number of free chain ends we may calculate the amount of fragments originating from such crosslinks at a given po:

Determination of Network Structure by Extraction and Random Degradation, 1 2205

At we > 0 there are N, ( 1 - we) network chains and 2 we Nc chain ends with a molecular mass MC,,/2. The number fraction 2 weNc/ (2Nc / f , ) , i. e. the fraction f , we of the 2 N c / f , crosslinks, bears a chain with a free end. This has the consequence that the elastically effective func- tionality decreases to a mean value f , (1 - we), at least for small we. Such crosslinks may also lead to the formation of fragments with a molecular mass fvMbr whereby one chain less is split than with the normal crosslinks. Therefore the generalized Eq. (42) has to be changed into Eq. (45) :

y w e = p ( l - jo,f”’---’ 2Mbr [ l I)] MC

(45)

For small we a similar consideration leads to Eq. (46) because each chain end is connected with ( f , - 1) network chains:

Z, random scissions occurring after crosslinking and before the start of the degradation experiment shorten the time t+ necessary for the dissolution of the network given by Eqs. (30) and (31). With Eq. (26) we find that t+ should be corrected by adding Z,/v,. Similarly, po should be corrected by adding Z,/Nc in the denominator of Eq. (17). Without this correction ofp,, the values Mi and [qli will be smaller than expected from the uncorrected degradation time. Then wi will be larger, even in a network that has been extracted before the degradation experiment. The influence of Z, on Mi and wi vanishes when the network is dissolved, if we use the values of p$ calculated from Eq. (25).

The quantity w, is increased further by the small number of rings consisting of two network chains since these may be split off even easier than fragments of the same size originating from crosslinks with chain ends; this correction of wi is small, however.

As shown by the considerations which lead to Eqs. (22) and (23), multiple splitting of network chains also produces small quantities of linear fragments of the network. The molecular masses of these fragments are distributed around 1/3, 1/4 etc. of the molecular mass M,, or - with M, distributions - of the molecular mass Msplit,c,n (see Eq. (21)). Their quantities therefore depend onp, and on their molecular masses.

For the parts with the molecular mass M,/3, the weight fraction is calculated by multiplying p2 by 1/3 because only the middle section is split off, while the chain ends remain connected to the network crosslinks. For non-uniform M, we find

Similarly, one finds wM /4,1in for non-uniform M, by multiplying j 3 by 1/2, because two of the four sections are split gff.

2206 M. Hoffmann

In nonideal networks, linear fragments result also when the network contains chains which are bound to the network on one side, and whose molecular masses are Mc, n 12.

The mean probability that such chains are split may be calculated with Eq. (14) to be (112). (ZINC). If such chains are present with the weight fraction we, then the number fraction which is split once is pl,,. Values of pl,e may be calculated with Eqs. (6) or (17) and Z/ (2NC) instead of ZINC. Withp,,,,., therefore, fragments with M,/4, and with fragments with M,/6 result. Similarly, with components with Mc,split,n/4 result. At we > 0 we have N, (1 - we) network chains and N , we 2 chains with one free, i. e. a number fraction 2 we of the original number N, and the weight fraction wl,e = 2we .pl,,. With non-uniform M,, e.g.:

For the type of molecule with Mc,split,n/4, therefore, the weight fraction obtained is larger than according to Eq. (49, with the result that, if M, and f , are known, the weight fraction we, which was not experimentally accessible from extraction for cases where scissions occurred after crosslinking, can be determined analytically by this procedure. If the chain length of the short chain branches is known, then similar equations may be derived for the fragments split off from such branches.

The elastically inactive loops may contain the weight fraction w, of the polymer. If w, differs from zero, this does not change the derived equations, since for f, = 4 the value of M, as defined above contains the material situated in loops.

4.2.2 Molecular mass distributions and GPC distributions of degradation products at dissolution of the networks for non-uniform

The average molecular masses and the degrees of branching of the degradation products with the lowest molecular masses were calculated in section 4.2.1 for networks with uniform f,. With these branched degradation products, the average molecular masses differed approximately by f , Mbr. But because there are molecular mass distributions for every type of fragment, the distributions overlap to form a unimodal overall distribution. The smallest branched fragment provides the largest weight fraction. In order to calculate the low molecular mass part of the overall distribution we assume that the M, values of the network have an exponential distribution with U = 1, i. e. with the coupling factor k = 1. The smallest branched fragments of the molecular mass Ml = f, * Mbr then have an exponential molecular mass distribution with the coupling factor k, = f,, at least for small degrees of splitting.

Regarding the branched degradation products at higher 1 - po, e. g. at 1 - p; , we find that repeated splitting of some network chains reduces the mean molecular mass of the branches. Some of them are smaller than the others. This reduces kl to k: . For the case of non-uniform M, at po = p i we obtain with Eqs. (22) and (24):

Determination of Network Structure by Extraction and Random Degradation, 1 2207

The molecules with the next higher molecular mass M2 have approximately double the k+ as the smallest ones and the next larger about three times the value of the smallest ones. But a further reduction of the k: results from the fact that the molecular mass Mc,split,n varies during the process of degradation. Eq. (22) shows that for f, = 4, Mc,,plit,n has been reduced from 2MC,, to 1,33 Mc,n when the network dissolves. As most of the soluble material is produced at about t+ , the value of k: is not reduced very much, for instance by about 2%. Such values of k: have to be used in Eq. (9), in order to calculate the distribution of the i-th type of fragments which will overlap with the distributions of the other fragment-types.

For practical evaluations of degradation experiments it is necessary to calculate GPC (gel permeation chromatography) elution curves of the degradation products. For this purpose we have to consider a further change of k: to kiteff because of the axial dispersion of the GPC columns (see Appendix 2). Furthermore, the molecular mass MApc in the maximum of a GPC elution curve is influenced by the branching of the molecules. Appendix 2 shows that in this case the maximum appears at an MbPC,app which depends on Mc,n and on the functionality f,:

If loops increase t+ to fzPp (see Eq. (31)), then the Mbpc,app will be reduced to M&pC,app,l. The latter may be calculated using At from Eq. (29) and dMb,,,,,/dt. The correction is small and reduces Mbpc,app by ca. 10% for f, = 4 and by ca. 6% for f, = 6 . A mean factor 1,89 instead of 2,05 should therefore be used in Eq. (52) for calculating M&pC,app,I. Eqs. (32) and (52) may be combined to yield an expression which givesf, as a function of measurable quantities:

Another determination of f , uses Eq. (31) and values of Mc,ex and we from extraction. Having evaluated f, we may use Eq. (52) (with the factor 1,89 instead of 2,05) for determining MC+. This value will be denominated Mc,deg. For networks which are inhomogeneously crosslinked or non-uniform with respect to f,, the quantities and molecular masses of the degradation products depend on the types of inhomogeneity and of the distribution off,. They may be calculated along similar lines.

The distribution cannot be calculated easily by the method given here for large molecular masses, but some conclusions may be drawn from the kinetics of the condensation of star-shaped molecules, of functionality fv, with one another 12). At the critical value p l = I/(& - I), M, and U become infinitely great. The distri- bution of the degradation products is then at its widest.

2208 M. Hoffmann

4.3 Intrinsic viscosity of un fractionated degradation products for non-uniform Mc,i

The effect of branching on the viscosity of solutions of the degraded material atp,' may be characterized by the branching parameter gb, (see Eq. (39)), which depends on the number m of branching points per molecule and on their functionality. For very large m and theta-solvents g,, approaches the following value1'):

gbr = vv 2(f, - l>(fv - 2)m (54)

Because of multiple scission of some network chains, the hydrodynamically effective functionalityf, of the branched fragments with the lowest molecular masses depends on degradation time. It is reduced to f ; in the same way as k, is reduced to k,+ (see Eq. (51)). Since the molecules with the next higher molecular mass are two stars attached to each other, their f ; values are approximately equal to those of the star-shaped molecules as two side chains of length 1 have nearly the same effect on gb, as one side chain of length 2. Tab. 1 contains values f i = k: for non-uniform M, and

Loops increase branching of the degradation products, but since w, is very small, we may neglect their influence on [q]&. At higher molecular masses and at p,' the branched fragments have m branching points of functionality f v and exhibit a third type of branching. Some of the crosslinks bear more than two unsplit network chains and act as branching points of long chain branching of the proliferative type with a functionality f ;eff of each of the m branching points. One may calculate the mean functionality f l e f f and [q]&:

Po = P i .

For f , = 3 we have 2Nc/3 crosslinks per cm3 and at p i a probability @,')'(1 - p,') for those crosslinks which bear two unsplit chains and are not branching points of the third type of branching; @,')3 is then the probability for crosslinks with three unsplit chains. Taking p,' = 1/( f , - 1) we find for fv = 3 an effective functionalityG,eff,3, keeping the number m of crosslinks constant and disregarding the fact that Mbr and Mc,unsplit, depend on time.

Similarly f;eff,4 = 237; f;eff,6 = 2,33 andfLeff,* = 2,20. A negligible amount of branching points forms rings of 2, 3, 4 and more unsplit chains. They increase f; insignificantly and therefore will be neglected. Molecules with the molecular mass Mi contain m = i branching points. Eq. (37) leads to Mi/i:

Now M&eff may be calculated with Eqs. (57) and (58) 2 Mi - 2Mi m=--

M,t, .fv MCfq.eff (57)

Determination of Network Structure by Extraction and Random Degradation, 1 2209

Then the values m$f = M/M,fq,eff andf;,,, may be inserted in Eq. (54) to yield gbr. This value will be not correct as it is based on the assumption that parts of the chain with the molecular mass M&,ff are linear, though they bear in fact side chains of molecular mass Mb: and possibly loops and short chain branches. The number of side chains with M$ per chain section with M,?, is given by Eq. (56) to be equal to (f, - 2). For a chain section with M&eff this number is (1, - 2). fv/flerf. The weight fraction of side chains with ML is (f, - 2) M&,nu/M& and reduces [q]& for very large molecular masses to a value given by Eq.,(59):

Values of [q]&/[qlMC,, are shown in Tab. 1. The effect of wsbr may be included in MC,, if the [q]-M-relation has been established with a polymer bearing short chain branches.

The preceding theoretical considerations lead to the result that extraction and degradation experiments may allow a quantitative determination of some structural parameters of networks which hitherto have not been analysable. This is possible for cases where the assumptions of the theory are valid: The extraction must be complete, and the degradation should be a random process. This means that chain scissions do not occur preferentially at crosslinks and do not proceed in the sense of a depoly- merisation starting at the chain end. Furthermore this theory was developed mainly for randomly crosslinked real networks with a single value of Mc and for those with an exponential M,-distribution with U = 1. The equations for other Mc distributions may also be easily derived. The degradation method has been developed for homogeneously and for inhomogeneously crosslinked real networks and for homo- geneously crosslinked networks with different functionalities of the crosslinks.

But the theory of extraction .is not yet adapted to all types of inhomogeneously crosslinked networks and to those model networks which are not produced by crosslinking a polymer with an exponential distribution of the molecular masses at random. Besides that, experimental errors influence the reliability of the results. This will be discussed in Part 2 of this series').

The random degradation method should also be valuable in the analytical deter- mination of the branching structure of soluble polymers.

Appendix I

The maximum of a GPC elution curve is found at a higher molecular mass than the maximum of the distribution function: When the distribution of Eq. (9) is plotted over a scale which is linear in M , it has a maximum at k .Mn. If one plots this distribution in the manner of a GPC elution curve over a scale which is linear in lnM, i. e. in such a way that the area above AlnM = In r is proportional to the mass of the molecules between Mand r . M , then the ordinate L of this distribution is given for small In r by

r M 1

Inr L = - j w ( M ) d M

Having carried out the integration, one can calculate the maximum of the GPC distribution using the condition dL/dlnM = 0 and obtains

2210 M. Hoffmann

(61)

A similar calculation yields Eq. (13) for the extracted material of a network.

Appendix I1

The GPC elution curve of a uniform polymer exhibits an apparent distribution with an apparent non-uniformity U, which reflects the influence of a characteristic of the apparatus, namely its axial dispersion. With good equipment and materials with high separation efficiency, U, = 0,M. For the following calculations we take U, = 0,07 which includes also the effect of a variation ofM,+,plit,n with time, which amounts to U, = 0,Ol. If U,is small, UGpc may then be calculated by adding this value U, to the non-uniformity U, of the molecular mass distribution.

The values k; and klef f of the smallest branched network fragments with molecular masses Mi' = fvM& can be seen in Tab. 1 . For fragments which are about twice as large (i. e. M2), the coupling factor is less than twice as large as k:, because of U, in accordance with Eq. (62).

If one does not determine Mdpc from the universal GPC calibration curve using the intrinsic viscosity of eluted fractions, the value of MApc in the maximum of the elution curve is lowered by branching of the degradation products to a value Mbpc,app, which will now be calculated. The weight fraction Wbr of branches in each molecule of type i has been discussed in section 4.2.1. Branching reduces the intrinsic viscosity to [tf]br,i (see Eq. (40)). The universal calibration curve of GPC giving M . [ q ] as a function of the elution volume V, is usually written in the form

I n ( M . [ q ] ) = A - B.V, (63)

From Eqs. (40) and (63) it follows that

MGpC,app,i = MGpC,i'gi:,(1!+4) = MGPC,i(l - Wbr,i)l'(l+u) (64)

With (I = 0,72 the exponent of (1 - Wbr) assumes a value of 0,58. But Eq. (64) describes the effect of branching on MGpc approximately correctly only if U = 0 14).

As U I 1/kteff, this effect is smaller than calculated by Eq. (64). Therefore 1 - (1 - w ~ ~ ) ~ * ~ * has been reduced by multiplying with 1 - 1/(3kceff). This is an approximation which is sufficient for our purpose and yields the values @;i;keff. For calculating the coil size distribu- tion of the molecules of type i we multiply w: in Eq. (44) with Eq. (9)

.exp - [ kifeff is calculated by inserting k: = i . ki' into Eq. (62). The distributions given by Eq. (65) are evaluated for several ratios M/M; and for all i between i = 1 and i = 13. At each M/M;

the sum 1 wz (M/M:) is calculated and plotted versus M/M:, in order to find the ratio of 13

i = l

Determination of Network Structure by Extraction and Random Degradation, 1 221 1

13 molecular masses M,,,,/M: at the maximum of the distribution. Then w: (M/M;) is di-

vided by ln(M/M: + 0,05) - In(M/M: - 0,05) and plotted versus In (M/MT) , in order to find the maximum of a GPC elution curve, i. e. MGpC,app/M:. The ratios MApC,app/Mc,n may be calculated using M:/Mc,n (Tab. 1). Increasing t to ca. 1,2 t + reduces M&C,app/Mc,n by about 15’7’0, whilst a diminuation of t f has nearly no effect. Reducing (I, to 0,04 changes M&C,app/Mc,n by ca. - 3%. Similar calculations have been performed with gbr,i = 1 yielding M,,/M, in the maximum of the molecular mass distribution plotted over a linear M-scale. Values of M,,,,/Mc,n are also shown in Tab. 1.

M. Hoffmann, Makromol. Chem. 183, 2213 (1982) M. Hoffmann, Makromol. Chem. 183, 2237 (1982) M. Hoffmann, M. Unbehend, Makromol. Chem. 58, 104 (1962) G. V. Schulz, Z. Physik. Chem., Abt. B: 43, 25 (1939) P. J. Flory, J . Am. Chem. SOC. 58, 1877 (1936); P. J . Flory, Chem. Revs. 39, 137 (1946) W. Kuhn, Kolloid-Z. 68, 2 (1934); E. Guth, H. Mark, Monatsh. Chem. 65, 93, 445 (1934) M. Hoffmann, Angew. Chem. 89, 773 (1977) M. Hoffmann, Rheol. Acta 6, 92, 377 (1967) F. Losch, “Tafeln hoherer Funktionen”, Teubner-Verlag, Stuttgart 1960 A. Charlesby, S. H. Pinner, Proc. R. SOC. London, Ser. A: 249, 367 (1959) M. Hoffmann, R. Kuhn, Makromol. Chem. 174, 149 (1973) P. J . Flory, J. Am. Chem. SOC. 69, 30 (1947); P. J. Flory “Principles of Polymer Chemistry”, Cornell University Press, Ithaca. N.Y. 1953. p. 374; W. H. Stockmayer, J . Chem. Phys. 12, 125 (1944) B. H. Zimm, W. H. Stockmayer, J. Chem. Phys. 17, 301 (1949) M. Hoffmann, H. Kromer, R. Kuhn “Polymeranalytik”, Band 1, Thieme Verlag, Stuttgart 1977, p. 379