Topology, thermodynamics and physical properties of polymer networks

Rheol. Acta 14, 113-1 26 (1975)

Po[ish Academy qf Sciences, Institute for Basic Problems of Technolo9y, Warsaw 1 (Poland)

Topology, thermodynamies and physieal properties of polymer networks II. Thermodynamieally most probable structures and related

physical behavior

A, Z i a b i c k i and W, K l o n o w s k i

With 8 fi9ures

Basing on the entropy considerations de- scribed in the former paper (1) one can look for topological structures which correspond to minimum free energy of the system. We assume here that enthalpy of the network system, AHnst is a function of crosslinking density, l, but is independent of the characteristics f and n~, Hence the Gibbs free energy of the network

AGùet = AHnet(l,s)- TASù~t(l,s,f,n~) [1]

depends on topology only through the entropy term, ASù~t. We will look for thermodynamically most probable structures assuming the total functionality of network junetions, s and crosslinking density, I to be constants. This corresponds to the maximizing of entropy ASne t with respect to the variables f (connectivity factor) and n~ (numbers of junctions with k single-bound chains) with additional conditions analyzed in the former paper (1)

ASn~t(f,n~,) = max; k = 0,1 .... ,s - 1

n~ = 2 N t / s q + 5) k [2]

k.n~ = 2N/( l + 1) k

f > l .

It has been shown (eq. [I. 34]*)) that the variables f and nj, do not appear simultaneously in all the contributions to erosslinking entropy ASùet:

ASnet(l,s, vp, f,n~) = ASais(l )

-k AScoùf(I,s, Op,n~) -4- ASe~(l,f). [3]

*) By eq. [I. n] we will understand eq. [n] appearing in Part I of the paper (ref. (1)).

138

(Received November 27, ¤972);

(revised December 28, ¤973)

Also the additional conditions in eq. [2] are separable in f and n~,

This makes possible separation of the entropy maximization problem into two problems:

ASel(f) = max [2a]

f > l

and

ASconf(n~) = max

n~ = 2N1/s(l + 1) « [2b]

n[ .k = 2N/(1 + 1) k

k = 0,1 ..... s - 1.

Both problems will be discussed separately and the resulting solution used for the calculation of most probable physical behavior of networks.

The most probable connectivity factorf

We will start with a simple case of Gaussian network chains subject to affine deformation A in the process of crosslinking. As shown in Part I (rel. (1)). the entropy ofelastic deformation of N ( I - 1)/(I + 1) double-bound chains reads

ASol = - 2~ k N [ ( 1 - 1)/(1 + 1)].(,~ ~ - 1 - l n ; o 2 ) .

[ 4 ]

At the same time, 2 is related to the connectivity factor f through eq. [I. 29]

, ~ 2 (h2~net/(h2~

= const, v p 2 1 3 M õ l / 3 ( l + 1 ) ( f / l ) 2/3. [5]

The constant in eq. [5] depends on the chemical structure of the polymer involved.

114 Rheologica Acta, Vol. 14~ No. 2 (1975)

~0 10 ~" lO 3 t.0 4

Differentiation of the elastic entropy contribution with respect of f yields

~ASe,/c~ f = (¢~AS,~/c~)~ 2) (d22/df)

= - ~ k N [ ( l - 1)/(I + 1)]

• (l -1 /22) (d22/«jO = 0. [6]

The solution of eq. [6] for finite f reads

fmùx = f (22~x) = cons t . - 3/2 1 l)p MÒ/2 (l -I- 1 ) - 3/2

ASob max = 0. [7]

In eqs. [5]-[7] vp denotes volume fraction of polymer in the network and M o - molecular weight of uncrosslinked (primary) macromolecules.

Since AS~~ is negatively definite, the optimum value indicated in eq. [7] is an absolute maximum. Eqs. [7] are valid only when fmù~ > I, i. e. when the crosslinking density 1, primary molecular weight Mo, and volume fraction of polymer, Vp satisfy the inequality

lVp(l + 1)- 3/2 Mò/2 ~ ( c o n s t . ) 3 / 2 . [ 8 ]

When the above inequality is not satisfied, fm~x = 1 and ASd . . . . = AS~~(f = 1) is negative. Formation of such networks however, is thermodynamically less probable than formation of a discontinuous system of relaxed molecular aggregates. Therefore the inequality [8] should be regarded as one of necessary (hut not suffi- cient) conditions of network formation. The result shown in eq. [7] means that the most probable connectivity factor f is one which nullifies the elastic entropy contribution. Conse- quently, the chains in a Gaussian network when formed are all relaxed

(h2)net = (hg) [9]

Fig. 1. Thermodynamically most probable connectivity factor fm«x as a function of crosslinking density, l. Molecular weights of primary (uncrosslinked) chains, M0 indicated

fmùx as calculated for a Gaussian network (eq. [7]) is a decreasing function of crosslinking density l, and an increasing function of polymer fraction Vp and primary molecular weight Mo. Numerical values Offma~ calculated from eq. [7] for vp - 1 (crosslinking in bulk state), Mo = 103, 104, 10 » and 10 ó with (const.)-3/• = 3.45 are plotted vs. l in fig. 1. It is evident that in wide fange of variables (covering practically the whole important range of I and M0) fm~x for crosslinking in bulk state is always greater than unity. The problem with the condition [8] arises however when crosslinking takes place in a dilute solution (% ~ 1). This is, why in dilute solutions separated aggregates replace a continuous network system.

We will examine validity of the conclusion about relaxed network chains (eq. [9]) and zero elastic entropy for non-Gaussian chains and/or non-affine deformations. To do this we will analyze the more general expression for ASe~ derived in Part I:

N ASel k~ {ln ,2 h = . [hj 7t( *)Ah*]

J

-- In [h 2 7J(h«) A h j]} [10]

where hj and h* are, respectively, the end-to- end distances of chains in uncrosslinked- and crosslinked states, and 7J(h) is the equilibrium distribution of end-to-end distances.

F o r non-affine deformations we can replace maximization with respect to a single variable f by maximization over local connectivity factors B for all chains, or over all end-to-end distances in the crosslinked state, h .2, j = 1 , 2 , . . , N ( l - 1)/ (l + 1) taken as independent variables• We are looking for solutions of the equation

elk«tl ~ et 2 ~ .• . ) ~--- m a x .

Ziabicki and Klonowski, Topology, thermodynamics and physical properties of polymer networks, /L 115

For Gaussian chains (cf. eq. [4]) any j-th term in the expression for AS~~ is negatively definite and the absolute maximum of AS¢~ corresponds to

h * = h j for allj [9a] ASyl = 0

what is equivalent to eq. [9]. The situation for non-Gaussian chains is

different. Since

Ah*/Ahj = h*/hj

the condition of maximization of ASel from eq. [10] with respect to independent variables h* can be written in the form:

OAS~I/Sh* = 3/h* + Oln~t'(h*)/ôh * = 0. [11]

Solutions of eq. [1l] depend on the form of the distribution function 7' and in general do not reduce to the result obtained for Gaussian chains (eq. [9]).

Assume the configuration distribution function for freely jointed chains of finite length in the form (cf. rel. (14)):

7J(h) = const. (l/h) L(h/mb) h

• e x p [ - b -1 ~ L(r/mb)dr] [12] o

where m and b represent, respectively, the number and the length of statistical chain segments, and L(h) is inverse Langevin function. Substitution of eq. [12] into eq. [11] yields the equation for optimum end-to-end distance

2h* + [L(h*/mb)] -1 dL(h*/mb)/dh*

- b -1 L(h*/mb) = 0 [11a]

which can be solved numerically for any given m and b. General analysis of eq. [11 a] indicates that its solution lies at h*2< mb 2 and the corresponding maximum entropy contribution is positive• For high enough m

~2 (hj)m~xmb2(1 - l / 5 m + 19/175m 2 + .-.). [13]

The corresponding deformation ratio 22, elastic entropy ASo~ per one chain and connectivity factor fj are

,~j2ma x , 2 2 = hj . . . . /mb = 1 - 1 / 5 m + . . .

B" . . . . = (const.)- 3/2

• vplMò/2(l + 1)-3/2(1 - 3/10m +. . . )

[7a]

ASel . . . . -~- k const.m-2(1 + .. ")

3k = 100m z (1 - 499/105m + .-.).

The number of statistical chain segments in the network chain, m is related directly to the molecular weight M and crosslinking density 1

m = tcM/b = xMo/b(l + 1)

where lc is molecular weight per unit length of the extended chain. It is evident that m incrcases with primary molecular weight of the uncrosslinked macromolecules, M0, and decreases with increasing crosslinking density l and increasing chain stiffness as characterized by the parameter b. For very long chains (m ~ oo) eqs. [7a] reduce to eqs. [7], i.e. to the result obtained for Gaussian chains.

As evident from eqs• [7a] the thermodynamically most probable state of a non-Gaussian network is one with chains not completely relaxed, but slightly contracted (2z< 1). The correction, proportional to the inverse number of statistical chain segments is usually very small, except for very short chains and/or very stift macromolecules. Such a situation can appear also at extremely high crosslinking densities.

In the literature concerning the theory of networks, opinions about the deformation of network chains in the process of crosslinking are diversified. Several authors (2, 3) imply that the average square end-to-end distance in an as-formed network is different from the relaxed, unperturbed value, characteristic of uncrosslinked chains

(hZ)ùe, 4= (h~)

without specifying the (h2)net values. On the other hand, Tobolsky (4) analyzed the model of a "non-penetrating network", i.e. the model equivalent to our "single network" with predetermined connectivity factor f = l, and arrived at (h2)net/(h 2) ratios different from unity. Our results (eqs. [7] and [7a]) result from the assumption of a variable connectivity factor, This indicates the role played by f in the determination of thermodynamic properties of permanent networks.

Distribution of junction types n~,

Maximization of entropy with respect to the distribution of junction types nj, can be performed for any given functionality s by solving

116 Rheolo9ica Acta, Vol. 14, No. 2 (1975)

eqs. [2bi. We will confine our analysis to Gaussian chains and tetrafunctional junctions a which can exist in 4 modifications: 4/0, 3/1, 2/2 and 1/3, i.e. junctions with k = 0,1,2 and 3 single-bound chains. Scheine of all these junction types was shown in Part I, fig. 1.

The configurational entropy of a tetrafunctional network as derived in Part I (eqs. [I. 16]- [I.18]) can be expressed through normalized b

4 4 variables nk --- nk/Znk in the following way:,

AS¢oùf = const .( l ,v v) + kN[1 /4 ( I + 1)1

• [ ( 1 - n3) ln(1 - n3)

+ (no + nl) ln(no + n l ) - - 2n3 lnn3

-- 2n2 lnn2 -- 2nl lnn l ] . [14]

Substitution into eq. [14] the conditions [2b], differentiation with respect to independent variables and simple rearrangements yield the following system of equations for nk:

n~ - n~(no + nl)no = 0" )

n4(no + nl) - n23n2(no + nl + 112) = 0

no + na -I- n2 + n3 -- I ---- 0 [15]

n 1 + 2 n 2 + 3 n 3 - - 4 / l = 0 .

Solution of the system [15] does not exist for l < 4/3• In this range of crosslinking densities no continuous system of chains connected by junctions can be obtained.

At l = 4/3 we have the solution:

n O ~--- g/1 ~-~ n 2 ~ 0 [15a]

n 3 = 1

which corresponds to the system of trimers, each consisting of 6 single-bound and 1 double- bound chains (fig. 2a)•

Also for higher /values: 4/3 < l < 1¢, solutions of eqs. [15] describe separate aggregates rather than a network• Position of the critical point however (gelation point, /~r) cannot be found from the thermodynamic considerations. From the analysis of the stochastic model of a "branching tree" (see Appendix) we obtain

1 = l~r "~> n » = 8 n o + 3nl

and

l~r = 1.705.

*) Here and everywhere below superscripts at n« denote powers of nk rather than functionality. So,

4 = (nk)4, etc. t/k

Fig. 2. Systems resulting in a tetrafunctional crosslinking process at asymptotic crosslinking densities. a) l = 4/3; n o = n 1 = n 2 = 0, n3 = 1 b) l = oo; n 1 = n z = n 3 = O,n o = l

The solution of eqs. [151 can be presented in the parametric form:

no = (1 + a) 1/2 [a 2 + (I + a)3/211/2/

• {a3(1 + a) 1/4 + [a 2 + (1 + a)3/2] 3/2}

nl = no" a

n2 = noaZ/(1 + a) 1/2 [15b]

n3 = noa3/(1 + a)~/4[a 2 + (1 + a)3/2] 1/2

l = 4{a3(1 + a) 1/4 + [a 2 + (1 + a)3/213/2}/

{3a3(1 + a) 1/4 + a[a 2 + (1 + a)3/2] 1/2

• [ 2 a + (I + a ) 1 / 2 ] } .

For infinitely high crosslinking densities l = o% a = 0 and the network is composed ofjunctions 4/0 alone (ideal network):

F/1 ~ H 2 ~ F/3 ~ 0

[15c] no = 1•

"Such an ideal system is shown in fig. 2b. Fig. 3 presents thermodynamically most prob-

able distribution of tetrafunctional junctions, n~ calculated from eqs. [15 bi and plotted vs. cross-

-8

-6

.q

0

50 40

Fig. 3. Thermodynamically most probable fractions of the individual junction types nk in a tetrafunctional network, plotted vs. crosslinking density, l.

Ziabicki and Klonowski, Topolo9y, thermodynamics and physical properties of polymer networks, II. 117

linking density 1. Similar analysis can be performed for any given functionality s. For a tetrafunctional network the fractions of 4/0 and 1/3 junctions are monotonical functions of crosslinking density l while those of 3/1 and 2/2 junctions exhibit maxima when plotted vs. I (fig. 3).

The number of elastieally effeetive network ehains, N«

In any network containing single-bound (free- end) chains, only part of the total number of chains participates in the transmission of elastic forces applied to the boundary of the system. Several formulas have been proposed for the calculation of the fraction of effective chains in the total number of network chains(7, 9, 10). We will derive Nee from the distribution of junction types n]~ and calculate N« values corresponding to the most probable junction distribution obtained in the preceding section.

It is obvious that from the total number of network chains, should be subtracted all single- bound chains

Nsingle ---~ 2N/(l + 1) = 2No,

This effect has been taken into account in the model analyzed by Flory (7). Not all double- bound chains are effective either. Their effectivity depends on the nature of junctions they connect and on the properties of the entire system.

We will consider first the distribution of double-bound chains with respect to the type of junctions placed on the two ends of each chain. If the absolute number ofjunctions with k single-bound chains is nL the number of unordered pairs of junctions with (k and j) single-bound chains will be

1.~t ~ 1) f o r j = k n~j = [16]

n~n} for j 4: k

j ,k = 0 ,1 , . . , s - 1.

To obtain the fractions of (k,j) pairs, or proba- bilities that a double-bound chain selected at random connects junctions with k and j single- bound chains, the values n~j from eq. [16] should be normalized to the total number of.pairs, to yield

{ ~~(,~-~~/~,~(~,~-~) ~o~»~ ~~k = [17]

2n~nS]~n]~(~,,n~- 1] forj :p k. /

For large enough numbers ofjunctions eq. [17] reduces to the more simple form

(fi],)z for j -- k h~, i = [17a]

2h[h) forj ~ k

where h~ = n~/~n~ are fractions of the indi-

vidual junction types. We will consider as "elastically effective"

those double-bound chains which connect two junctions each of which has not less than two tracts leading to infinity (i.e. to the boundary of the system). This definition of an "effective chain" seems to be identical with that proposed by Scanlan (9), Mullins and Thomas (10) but formulas for N« derived by those authors are different from our results.

We will write the number of elasticaUy effective chains in the form

Nee = No(1- 1 ) ~ ~,Qjkfi[j. [18] k>=j

No(1 - 1) represents the total number of double- bound chains and Qjk are effectivity coefficients for the individual classes of chains. The coefficients Qjk are determined by two factors - the nature of junctions attached to the ends of the considered chain and the probability that the tracts issuing from these terminal junctions will lead to the boundary of the system. Conse- quently Qjk can be split into two factors

Qjk = Cik Wk(p) [18a]

where the first factor C j~ depends only on the type of junctions directly attached to the ends of the chain, and pik is a numerical factor de- pendent on the probability p that an arbitrary tract in the network leads to infinity. The probability p has been derived in the Appendix.

If all tracts in the network lead to infinity (pik= 1) all chains connecting junctions with 3 or more double-bound chains (i. e. junctions with k ~ s - 3 single-bound chains) will behave as elastically effective. Therefore we can write

C j k = l f o r j < s - 3 a n d k = < s - 3 . [19a]

On the other hand, chains connecting one junction with (s - 3) or less single-bound chains and another junction with exactly (s - 2) single- bound chains (k < s - 3, j = s - 2) contribute only in one half to the number of elastically effective chains. This is illustrated for a tetra-

118 Rheologica Acta, Vol. 14, No. 2 (1975)

functional ne twork in fig. 4a. Two chains (2,0) are equivalent to one effective chain only

2 C 2 ' ° = 1; C 2 ' ° = 1/2.

In general we have

C i k = l / 2 f o r k _ < s - 3 a n d j = s - 2 . [19b]

Chains (s - 2, s - 2) connecting two junctions, each having ( s - 2) s ingle-bound chains are ineffective. This can be easily demons t ra t ed on the te t rafunct ional example shown in fig. 4b. The series of two chain is equivalent chain.

2C 2'° + C 2'2 = I •

(2,0) chains and one (2,2) to one elastically effective

Subst i tu t ion of the former result: C 2 ' ° = 1/2, yields

C 2'2 = 0 .

In general we have

C j«=O f o r . j = k = s - 2 . [19c]

The same is true for chains which lead to junct ions having ( s - 1) s ingle-bound chains. An example is shown in fig. 4c. Genera l ly

C ~k=O f o r k = s - 1, a n y j [19d l

or j = s - 1 , a n y k .

Tak ing into account symmet ry of the effectivity coefficients C ~« = C kj we can present the matr ix C j« in the fo rm

k = ~ ' = 0 1 ... s - 3 s - 2 s - 1

0 1 1 ... 1 1/2 0

1 1 1 ... 1 1/2 0

C ~k . . . . . . . . . . . . . . . . . . . . . .

s - 3 1 I ... 1 1/2 0

s - 2 1/2 1/2 ... 1/2 0 0

s - 1 0 0 ... 0 0 0

Consider now factors W k. The probabi l i ty that in a group of r doub le -bound chains n chains lead to infinity is

where p is probabi l i ty that a single t rack leads to infinity (cf. the Appendix). The probabi l i ty that in a g roup of r chains 2 or more lead to infinity is

Fig. 4. Elastic equivalence of various types of double- bound chains in a tetrafunctional network. a) 2. (2,0) ~ 1-(0,0) b) 2. (2,0) + 1. (2,2) ~ 1. (0,0) c) t . (3,0) + 1. (0,0) ~ 1. (1,0)

qr= P(n >=2)= 1 - P ( 0 ) - P ( 1 )

= 1 - (1 - p)~- ' [1 + p(r - 1)]. [22]

An s-functional junct ion with k s ingle-bound chains provides r = (s - k - 1) doub le -bound chains, two of which should lead to infinity if the junct ion is to terminate an elastically effective chain. Therefore the factor pik for a chain connect ing s-functional junct ions with k and j free ends reads

pik -= p k j = qs-k- 1 qs- j - 1, for k,j < s - 2. [23a]

For k = s - 2 the probabi l i ty qr = 1 as calculated f rom eq. [22] is equal to zero. However , chains of the type (k, s - 2) are not complete ly ineffective; the condi t ion of m i n i m u m two tracts leading to infinity cannot be met by the junct ion (s - 2) but can be t ransmit ted to the further junct ions along the chain having enough (more than 2) doub le -bound chains. More than that, if the junct ion ( s - 2) is followed by a sequence of m junct ions ( s - 2) the condi t ion of two t racks can be satisfied by a junct ion of the (m + 1)-st generation. This is i l lustrated schematical ly in fig. 5 for a te t rafunct ional network. Therefore for a ne twork chain which connects a junc t ion with k < s - 2 s ingle-bound chains and a junc t ion with j = s - 2 we will obta in

PS-2 'k=q*qs_k_l , f o r k < s - 2 [23b]

where q* is a weighted average of probabil i t ies qr for var ious junct ions possibly following the junct ion (s - 2)

~-~ /~~ -s -, [243 q* = ~ nk qs-k- t nk.

k ~ O k k4~s--2 I k~ßs~2

Ziabicki and Klonowski, Topology, thermodynamics and physical properties of polymer networks, H. 119

GENERATION 0 1 2 3 4

1 1

T ~ ~ - - ~ Fig. 5. Transmission of the "two-tract condition" along the chain containing sequences of junctions (s - 2). Tetrafunctional networks, a) single junction 2/2. The condition transmitted to the

second generation and satisfied by junction 4/0 b) single junction 2/2. The condition transmitted tothe

second generation and dissatisfied by junction 1/3 c) sequence of 3junctions 2/2. The eondition transmitted

to the fourth generation and satisfied by junction 3/1

The summation in eq. [24] exeludes junctions with k = s - 2 because the appearance of such junctions in a sequence simply shifts the condition to further generations.

Multiplying for every pair of indices q, k) the factors C Jk from eq. [20] by pjk from eqs. [23a, b] one obtains symmetrical matrix of effectivity coefficients Qjk for an s-functional network in the form

Q~=

-k = / j = 0 1 . . . s - 3

0 q~-a q s - l q s - 2 ... q s - l q 2

1 q2 2 "" qs-2q2

s - - 3 q~

s - - 2

s - 1

The number of effective chains, N« can be found for any functionality s flora eqs. [18] and [25]. For tetrafunctional networks combination of eqs. [18] and [26] yields

N « = «o( l - 1)p4[nl + no(3 - 2p)] 2

• (no + nx + n2 + na)/(no + H1 -~- H3). [27] The probability p that a single tract leads to infinity is derived in the Appendix. For tetrafunctional networks

p = }(1 + nl /4no)

- ½[(1 + 3nl /4no) 2 + n3/no] ~/2. [2S]

Eq. [27] differs from the relations for effective chains derived by previous authors. In the mode! of Flory (7) only single-bound chains are considered as ineffective. In our notation this cän be written as

N « = N - 2 N o = N0(Z - ~) . [ 2 V a ]

This relation is obtained from out eq. [27] when n o = l and n l = n 2 = n 3 = 0 , i.e. for an ideal network. On the other hand, the expression found by Scanlan (9) and Mullins and Thomas (10)

N « = Nol(no + 3na/4) [27b]

cannot be easily obtained from our eq. [27] except for 1 ---, 0% although both analyses (9, 10) started from the effectivity criterion identical as in our paper (minimum two tracts leading to infinity from each terminal junction of the effective chain). The difference between the results of Scanlan,.Mullins and Thomas on one hand and ours on the other, seems to arise from

s- -2 s--1-

½q~-i q* 0

½ qs- 2 q* 0

½q2q* 0

0 0

0

[25]

For a tetrafunctional network (s = 4) the matrix Q reduces to

Q• = p4. [ (3 - 2p) z (3 - 2p)

1

½ ( 3 - - 2 p ) [ n l + n 0 ( 3 - 2 p ) ] / ( n o + n l + n 3 ) 0]

½ [nl + no(3 -- 2p)]/(no + nl + n3) ~] [26] 0

0


the fact that the previous authors considered distribution of effective (or ineffective) network junctions while we studied distribution of network chains. The latter approach seems to be more appropriate for the effectivity problem involved. The differences between our eq. [27] and the equations of Scanlan, Mullins and Thomas concern any distribution of network junctions n» Another source of differences is associated with the form of this distribution which in Scanlan, Mullins and Thomas papers is assumed binomial while in this paper is obtained from thermodynamic considerations.

It is evident that in our expression for effective chains (eq. [27]) appear not only fractions of junctions with 4 and 3 double-bound chains (no and nl) but also those ofjunctions with 2 and 1 double-bound chains (n2 and n3). Although the absence of junetions with 4 and 3 double- bound chains excludes the presence of elastically effective chains

n o + nl = 0 » N e r = 0

the fractions of apparently "ineffective" junctions n2 and n3 do affect the values of Ne.

P,~

1.0

.6

f

2 5 10 20 50 100 200

N~dNo

1000

lo0

10

Fig. 6. The relative number of elasticaUy effective chains (Ner~No), the fraction of effective chains q~ = N«/N«oubl~ and the probabiiity that a single tract in the system leads to infinity, p, plotted vs. crosslinking density, l. Tetrafunctional network with thermodynamically most probable structure

Numerical calculations have been performed for tetrafunctional networks. Fig. 6 presents the relative number of effective chains calculated from eq. [27] with most probable fractions of junctions nk as derived in the preceding section (eq. [15b]). It is evident that the number of effective chains per orte primary macromolecule, Nef /No rapidly increases with crosslinking den-

sity parameter l, at high l's being nearly proportional to I. The effectivity coefficient

(p = Nef/Ndoub|e = Ner~No(1 - 1)

monotonically increases in the range 1.7 < l < 50 and practically levels oft (~o ~ 1) at higher densities of crosslinking. Similarly behaves the probability p that a single chain leads to infinity, (eq. [A7]) but it levels oft already near l - 7. The critical crosslinking density ("gel point ') lù

lc~= l(p = O, N e = O )

appears in our case near l = 1.705.

Modulus of elasticity and elastic behavior

Stress tensor in a system of N« Gaussian chains can be expressed through the average dyadic, formed of the end-to-end vectors h of the individual chains (5)

p =- k T ( N « / V ) [ 3 ( h h r ) / ( h Z o ) - I] [28]

where Vis volume of the system, (hg) - average square end-to-end distance for uncrosslinked, unperturbed chains, and I - unit tensor.

Consider a Gaussian network in two states: i., as-formed and ii. subjected to macroscopic deformation with displacement gradient tensor A. As shown in Part I, formation of a Gaussian network with connectivity factor f is accom- panied by spherically symmetrical deformation of all vectors h and the corresponding extension of the end-to-end distances

(h2)n~t = 22(f) (hg) . [29]

Assuming that the subsequent macroscopic deformation (transition from the state i. to ii.) is equivalent to an affine deformation of all vectors h

hde f = Ahne t [30]

and the distribution of vectors h in undeformed network (state i.) is isotropic

(hhT)net ~__ 1 (h2)net I [31]

one obtains

(hhT)de f = ½ (h2)net A A r [32]

and

p(A) = k T(Nef /V) [22 A A T - I ] . [33]

It is evident that the elastic behavior depends on two factors:

Ziabicki and Klonowski, 7bpolo9y, thermodynamics and physical properties of polymer networks, II. 121

1. concentration of elastically effective chains ( N e ~ V ) related to shear modulus

G = (Ne~V) k T, [343

2. deformation of network chains introduced in the process of crosslinking, 22o.

The effect of network topology on the number of elastically effective chains was discussed above. N« (and therefore G) increases with crosslinking density 1 and in the fange of small and moderate l's depends on the distribution of junction types nL Numerical data for tetrafunctional networks have been shown in fig. 6.

The other problem concerns the coefficient 2o2. For Gaussian networks with variable connectivity the most probable f is such that"

~o2(fmax) = I [35]

and

p(A) = k T ( N « / V ) [ A A T - I ] . [36]

Assumption of a constant connectivity factor f0 (e. g. fo = I like in the paper by Tobolsky (4)) leads to a different behavior

p(A) = k T ( N « / V ) [ c o n s t . f 2 / 3 A A T - I] [37]

where also macroseopically undeformed state (A = I) is not relaxed but involves some internal pressure

p(I) = k T(N~dV) [const. f2o/3 _ 1] I . [38]

Eqs. [33]-[38] indicate that network connectivity plays very important role in the determination of elastic behavior.

Free energy of swelling

The other macroscopic physical behavior sensitive to network structure is swelling. To calculate the effects of connectivity factor f and junction type distribution nE on the free energy of swelling we will follow the general lines of reasoning proposed by Flory (6).

Consjder a permanent network crosslinked in a dry state (Vp = 1) consisting of N chains and characterized by variables (l,s, fn[) . The process of swelling is equivalent to mixing of the network with Ns molecules of solvent to reach the volume fraction of polymer Vp < 1 without any change of structure.

Considering the cycle of states analyzed by Flory (6) it is easy to find that the Gibbs free energy of swelling can be expressed by the following equation

AG~,~ = AH~w - TAS~w = AH~w

- T[ASoo«(Vp) - ASeoùf(Vp = 1)

+ ASmi . . . . (vp) + AS~1(2 = 20v~ 1/3)

- ASel0~ = )~o)] [39]

where AH~w is enthalpy of polymer-solvent interactions, AS~onf(Vv) - configuration entropy of chain ends being connected in the system with polymer fraction vp, ASmi . . . . - - entropy of mixing of N network chains with Ns molecules of solvent in an uncrosslinked stare, and A S~1(2) - entropy of elastic deformation of network chains. 2o is the coefficient of molecular deformation accompanying crosslinking in the bulk state (vp = 1).

From the Flory-Huggins theory of mixing (7) we have:

ASmi . . . . = - k IN, ln(1 - vp) + N lnvp]. [40]

As shown in out former paper (1) (eqs. [I.20] and [I.24])

ASco,e = k N- const.(l, s, n~)

+ 2 k N l [ ( s - l)/s(l + 1)] lnvp [4¤]

and for Gaussian chains

AS¢,(2) = - ~ k N « ( 2 2 - 1 -ln2X). [42]

From the combination of eqs. [39]-[42] we obtain

AGsw = AHsw + 3 k T N « 2 2 ( v ~ 2 / » - 1)

+ k T [N + N « - 2 N l(s - 1 )/s (l + 1)] In vp

+ k T N s ln(1 - vp). [43]

Since the free energy of mixing dry network with Ns solvent molecules is

AGmi . . . . t -- AH, w + k TN~ ln(l - vp) [44]

it is a common practice to separate the "elastic" part of the free energy of swelling

AGel = AGsw- AGmi . . . . t [45]

which can be presented in the general form

A Ger = A . ~ k T N « ( v ; 2/3 _ 1) + B - k T N In vv.

[46]

For the network with structure (l,s, fn~) the constants A and B (cf. eqs. [43]-[46]) read:

A = 2o2(./); B = 1 + Nef /N - 21(s - 1)/s(l + 1);

N « = N e ( N , l, s, nT,). [47] 9


For networks with variable connectivity the most probable structure derived from eqs. [2] leads to the following values of A ("front factor"):

A = 1 for Gaussian chains

A = 1 - 1/5m + ... for Non-Gaussian chains.

[48]

On the other hand, assumption of a constant connectivity factor, f = fo yields

A =fo 2la ( r 2 ) / ( h 2) = A(l) 5~ 1. [49]

(r 2) is the average square distance between nearest in space network junctions and (hg) - the average square end-to-end distance for uncrosslinked network chains.

The coefficient B is a complex function of network structure. Numerical Values of B for a tetrafunctional network with thermodynamically most probable structure (Ne from eq. [27]) are plotted vs. crosslinking density I in fig. 7. B varies in a wide fange of values between 0.01 and 0.5 approaching asymptotical value B = 0.5 at l - , oo and exhibiting a minimum near l = 2.05. The function B(1) levels oft only above I = 200. More stable characteristic is provided by the product (BNtot/Nef) which above I = 2 lies within the range (0.27-0.51) and practically levels oft already above l = 10.

B

,8

.7

.6

.5 - -

.4

.3

,2

.1

0.0

B N~o~;Nef

2 5 10 20 50 100 200

Fig. 7. Thermodynamically most probable swelling coefficient B and reduced swelling coefficient (B Ntot/Nef) vs. crosslinking density l. Tetrafunctional network

The asymptotic form of the swelling equation for an s-functional network with most probable topology at l ~ oo assumes the form

A Ge1 = ?z k T N(v~ 2/3 _ _ l) -[- (2 k T N / s ) In vv [50]

identical with that derived by Flory (6).

Diseussion

The present two papers have been concerned with several problems. The first one is effect of structural characteristics on the thermodynamics and physical behavior of polymer networks. The network model considered does not differ physically from earlier models analysed by James and Guth (2), Staverman (3), Flory (6), Hermans (8) and others. The difference lies in mathematical treatment: some additional structural eharacteristics - the connectivity factor f and junction type distribution n~ - have been introduced explicitly into the entropy calculations. The entropy of crosslinking has been calculated using a modified configuration lattice method what made possible consideration of non-Gaussian chains and non-affine deformations.

It has been shown that the topological characteristics strongly affect thermodynamic and physical behavior of networks. Beside the usually analysed parameters s (junction functionality) and l (crosslinking density), an important role is played by the connectivity factor f related to chain deformation in the process of crosslinking and fractions of network junctions with k = 0,1 .... ,s - 1 single-bound chains, nL The number of elastically effective chains, modulus of elasticity and swelling parameters all depend on f and n~.

The other problem is thermodynamically most probable structure and related physical properties. Formally speaking, permanent networks are irreversible systems by definition and no thermodynamic equilibria can be reached. However, any chemical crosslink is in fact reversible, exhibiting finite, though often very high, dissociation energy. Therefore we will consider permanent networks as a limitin9 ca se of reversible systems at high dissociation energy, what justifies the minimization of free energy and at the same time guarantees constancy of the number of crosslinks.

Mathematically speaking, we consider hefe a minimum of free energy G with respect to topological variables x l , x 2 . . . . . x"

G(x l , x 2 . . . . . X n) = min . [51]

It is generally true for any function of n variables that optimization with respect to all n variables yields better results (i.e. higher maxima, or deeper minima) than optimization with respect

Ziabicki and Klonowski, Topolo9y, thermodynamics and physical properties of polymer networks, H. 123

to m < n variables with the remaining ( n - m) m+ 1, X~*+2 assumed constants variables x o ...

1 2 n G ( x m i n , X m i n , - - . , X m i n )

~ , ' 1 2 m . m + 1 = [J[Xmin, Xmin , . . . ,Xmin ,XO . . . . . X~) . [52]

Consequently, introduction of additional variables (subject to variational analysis) will always lead to solutions thermodynamically more probable (ones with lower free energies) than optimization of structures with lesser number of variables. If only the thermodynamic approach is justified, our results will be more probable than those obtained for network models assuming a priori some specific values of the characteristics f and n~.

The important result of such optimization is connectivity factor fmax such that maximizes the entropy of elastic deformation experienced by polymer chains in the process of network formation (crosslinking). Gaussian chains are completely- and Non-Gaussian chains nearly completely relaxed when formed in the crosslinking process; the average square end-to-end distances of as-formed network chains are equal (or nearly equal) to those for uncrosslinked chains in solution

( h 2 ) n e t = (ho z) for Gaussian chains [53a]

ASel ( fmax) = 0

and

(h2)ù~t = (hg)(1 - l/Sm +.. .) = @2) +O(1/m)

ASom(fmùx) = (3 k/lOOm 2) [53b]

[1 -499/105m + --.] = 0 +O(1/m 2)

(where m » 1) for Non-Gaussian chäins.

If, on the other hand, the connectivity factor is not subject to optimization but assumed constant ( f = f o), optimization with respect to the remaining variables yields (for Gaussian chains)

(h2)net = const. (h2) f 2/3 ~ 1 [53c]

ASt, = ASe~(fo) < O.

The resulting structure exhibits higher free energy than that corresponding to variable f, the difference being k TAS~t(fo).

In table 1 are collected theoretical results obtained by various authors for the fundamental characteristics of permanent networks: the fraction of elastically effective chains, (Nef/N) reläted to shear modulus G, and the coefficients A and B

in the equation of swelling. A = (h2)net/(h 2) is also the characteristic describing chain deformation in the process of crosslinking.

As evident from table 1 our predictions for thermodynamically most probable structures in the limit of infinite I reduce to the results of Flory (6).

The coefficient A (front factor) equal to unity derived in our theory from the minimization of free energy with respect to connectivity factor f, existed in earlier theories (6) as an assumption. On the other hand, the results for networks with predetermined connectivity f = f o (eq. [53c]) are identical with those obtained by Tobolsky (4) for f0 = 1. Front factor A consid- erably different from unity appears to be thermodynamically improbable for long, flexible, Gaussian chains. It can result, however in crosslinked systems composed of very short, rigid chains or when network structure is controlled by factors other than thermodynamics (e.g. kinetics of crosslinking).

Our calculations of the entropy of crosslinking (ref. (1)) based on an independent method (modified configuration lattice approach) con- firm the results obtained by Flory (6) and yield the same valucs of the coefficients A and B" in the limit of infinitc crosslinking densities.

We do not claim out results to represent exactly the actual behavior of real networks, and that for two reasons:

i. it is not established that actual network structures are controlled solely (or mainly) by thermodynamic factors,

ii. in our network model, many important structural features (first of all, chain entangle- ments and molecular weight distribution of network chains) have been neglected.

What we aimed in this and the preceding paper (1) was to apply a "thermodynamic test" to the network model which, oversimplified as it is, became a widely used tool in the inter- pretation of rubber elasticity. We believe also that comparison of the predictions for thermodynamically most probable structures with experimental data and predictions based on purely kinetic considerations can throw some light upon the mechanism of network formation.

Added in proof :

The analysis involved in this and the preceding paper(1) involves an extension of the Flory theory of crosslinking (6). It is the present

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Ziabicki and Klonowski, Topology, thermodynamics and physical properties (?f polymer networks, Il. 125

feeling of the authors that the method of cal- culating c r o s s l i n k i n g e n t r o p y b a s e d on the count ing of independent arrangements of network chains is not justified for permanent ly crosslinked systems. Pe rmanen t networks actu- ally do not involve the degrees of freedom (rotat ional mot ions o f end-to-end vectors) which yield the volume factor 4~ h~ Ahj in crosslinking entropy. Consequently, we think that the treatment presented above can possibly be still applied to reversible networks but is incorrect for permanent ly crosslinked systems. The problem of front factor A and the swelling factor B in rubber elasticity seems to remain still open for discussion.

Appendix

The probability p d~at a tract in the network leads to infinity

To find the probability p that a network chain issuing from an arbitrary junction leads to infinity, we will start from the stochastic model of a "branching tree" (11). This model, widely used in the theory of crosslinking (cf. 7, 12, 13) does not describe network systems sensu stricto but infinitely branched molecules. We will use this model in a modified form and that only for the calculation of the probability p. All the characteristics of network structure - ( f , fiT,) are derived independently from thermodynamic considerations without any assmnptions involved in the theory of branching processes.

Consider transitions from õne generation of network junctions to the next generation. In an s-functional crosslinked system the probability generating function F~(~9) for the transition from the zero-th to the first generation reads:

s

F~ (0) = ~~0fi], O (~- k). [A 1J

~~, the probability that a junction in the zero-th generation will have ( s - k) descendants in the first generation, equals to the fraction of s-functional junctions with k free ends. Similarly, the generating function for the transition from the first to the second generation is

s ~ l s - - l -~

F](O) : k~o(S- k)h~O ('-k- ~) k~=o(S - k)B~ . [A 2]

Typical transitions for a tetrafunctional network are shown in fig. 8.

In a couple of papers concerned with rubber elasticity(9, 10), probability generating fnnctions similar to those given by eqs. [A I] and [A 2] were discussed. The distribution of junction types ~~ was assumed to be binomial while our eqs. [A 1, A 2] allow for a general case with arbitrary distribution fi~. In numerical calculations we substitute into the generating functions the fractions of various junctions fi~ obtained from thermodynamic optimization (cf. eqs. [15bi in this paper).

1

0/4 1/3 2 / 2 3/1 4 /0

2

o b 1/3 2 / 2 3/1 4/0

Fig. 8. Stochastic model of a brancbing tree for tetrafunctional systems, a) transition from the zero-th to the first generation

(probability generating function F4(O)) b) transition from the first to the second generation

(probability generating function F~'(0))

According to the theory of branching processes (11) the probability of extinction of a single tract in infinity

e = 1 - p [A 3]

is the smallest root of the equation

e = F] (e) [A 4]

which satisfies the condition

0 <= e <= 1. [A 53

Eqs. [ A 2 ] - [ A 5 ] determine the probability p for networks with any functionality s, and any distribution of junction types, fi~, in the approximation offered by the "branching tree" model. For a tetrafunctional network (s = 4) the combination of eqs. [A2], and [A 6] leads to the equation

e = (4noe 3 + 3nie z + 2n2e + n3)/ [A 6]

(4no + 3nl + 2n 2 + n3)

which, solved for e yields

e = Ü-(1 + 3nl/4no) [17 ] • {[1 + n3/no(l + 3n~/4no)Z] ~'2 - 1}

and ultimately

p = 3 ( 1 + nl/4no) [A 8] 1 2 [(1 + 3nl/4no) 2 + n3/no] 1~2

where nk (k = 0,1,2, 3) is an abbreviation for fi2.

Acknowledgement

One of the authos (A. Z.) is greatly indebted to Pro- fessors P. J. Flory, A. J. Staverman and K. Du~ek for their eomments concerning the problems dealt with in this paper.

Summary

The molecular model of an s-functional network has been completed by introducing additional structural variables: connectivity factor f (related to chain defor-


mation in the course of crosslinking), and fractions n~ of s-functional junctions with k (k=O,1 ..... s - l ) single-bound chains. Using the crosslinking entropy derived in.the preceding paper (1) thermodynamically most probable structures (i.e. ones corresponding to minimum ffee energy) have been calculäted and discussed.

It has been shown thät for Gaussian ehains, crosslinking does not change the average square end-to-end distance and the as-formed network chains are relaxed

(h2)net = (hg).

This result has been shown to be approximately valid also for non-Gaussian chains provided that they are not too short and not too stift.

The basic physical properties of networks with thermodynamically most probable structure: shear modulus G, the number of elastically effective chains, and the swelling constants A and B have been calculated and discussed. The constant A ("front factor") in the equation of elasticity is equäl to unity (for Gaussian chains) or slightly less then unity for non- Gaussian chains. The other swelling constant B and the fraction of elastically effective chains, N«f/N äre complex functions of crosslinking density and network structure. The expression for the fräction of elastically effective chäins Noe/N differs from the formulas proposed by Flory (7), Scanlan (9), Mullins and Thoma} (10) and other authors. In the limit of infinite crosslinking density (l --+, co) thermodynamically most probable values: A = 1, B = 2/sNef/N = 1 are identical with those derived by Flory (6, 7).

Rejor«nces

1) Ziabieki, A. and W. Klonowski, Rheol. Acta 14,105 (1975).

2) James, H. M. and E. Guth, J. Chem. Phys. 21, 1048 (1953).

3) Staverman, A. J., in: S. Flügge (Ed.), Handbuch der Physik, vol. 13 (Berlin-Göttingen-Heidelberg 1962).

4) Tobolsky, A, V., Thesis, Princeton, 1944. - D. Katz and A. V. Tobolsky, Polymer 4, 417 (1963).

5) Ziabicki, A., Pure and Appl. Chem. 26, 481 (1971).

6) Flor T, P. J., J. Chem. Phys. 18, 108 (1950). 7) Flory, P. J., Principles of Polymer Chemistry

(Ithaca N.Y. 1953). 8) Hermans, J. J., J. Polymer Sci. 59, 191 (1962). 9) Scanlan, J., J. Polymer Sci. 43, 501 (1960).

10) Mullins, L. and A. G. Thomas, J. Polymer Sci. 43, 13 (1960).

11) Rosenblatt, M., Random Processes (New York 1962).

12) Stockmayer, W. H., J. Chem. Phys. 11, 45 (1943). 13) Gordon, M., T. C. Ward, and R. S. Whitney, in:

A. J. Chömpff(Ed.), Polymer Networks. Structural and Mechanical Properties (New York 1971).

14) Flory, P. J., Statistical Meehanics of Chain Molecules (New York 1969) p. 321.

Authors' address:

A. Ziabicki and W. Klonowski Institute for Basic Problems of Technology Polish Academy of Sciences PL-00-049 Warsaw 1 (Poland)

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Topology, thermodynamics and physical properties of polymer networks