Critical behavior of trifunctional randomly branched polycyanurates

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Critical behavior of trifunctional randomly branched polycyanurates

Jörg Bauer, Walther Burchard

To cite this version:

Jörg Bauer, Walther Burchard. Critical behavior of trifunctional randomly branched polycyanu- rates. Journal de Physique II, EDP Sciences, 1992, 2 (5), pp.1053-1063. �10.1051/jp2:1992185�.

�jpa-00247691�

Classification Physics Abstracts

82.35 05.50

Critical behavior of trifunctional randomly branched

polycyanurates

J6rg Bauer (*) and Walther Burchard

Institute of Macromolecular Chemistry, University of Freiburg, W-7800 Freiburg, Germany (Received 9 January 1992, accepted 31January 1992)

Abstract.-The behavior _near the gelation threshold of trifunctional randomly branched

polycyanurates is studied by static and dynamic light scattering. By static measurements the critical exponents y, a and _{v were} obtained, which describe the divergence of the weight average (M~) and the characteristic (M* ) molecular weights and the radius of gyration (R~) respectively.

All these independently measured exponents together with _T, describing the power law behavior of the molecular weight distribution and measured by size exclusion chromatography coupled

with light scattering, confirm the predictions of the three-dimensional percolation theory.

Furthermore, it _was shown by dynamic light scattering that the power law behavior _{over some} decades in time of the time autocorrelation function and the divergence of the _mean relaxation time _are characteristics of the gelpoint. The development with increasing reaction time of the time

correlation function of the gelling system from the pregel through ^the gel-point into the gel state was analyzed quantitatively by a hybride of _a stretched exponential and _a power law function. It

was shown that the basic relations between _{y, ~, v} and _{T are} fulfilled and should be universally

valid. On the other side, the relations, which link the characteristic exponents of the time correlation function _at the gelpoint to the basic exponents, gave incoherent predictions. In that

way, one can assume that their universal validity ^is limited, since _some additional assumptions _on

the local dynamics _are needed.

Introduction.

In _a previous _paper [Ii _we introduced polycyanurates, the polycyclotrimerization products of difunctional aromatic cyanic acid esters, as a model for trifunctional, randomly branched

polymers [2, 3]. As _a first result the scaling relations

R~ M$ R~ M$ with p'

=

0.52 _± 0.03 (1)

for the radius of gyration R~ m ((s~) )~'~ and the hydrodynamic radius R~

m ( I/R )

~

)~ _were

(*) Permanent address Fraunhofer Institute of Applied Materials Science, O-1530 Teltow-Seehof, Gernlany.

obtained by static and dynamic light scattering. Furthermore, the predicted _power law behavior of the weight ^fraction molecular weight distribution [4]

w (M ) M ^~ f (M/M *) with _T

=

2.22 _± 0.02 (2)

was found using size exclusion chromatography coupled with _a low angle light scattering detector, where f is _a cutoff function and M* _a characteristic cutoff molecular weight. The exponent p' contains informations _on both the molecular weight distribution and the solvent

quality and is related to p for the molecular uniform clusters by p = p'(3 T), which gives

p = 0.4I and _a fractal dimension D

=

I/p

=

2.44. The excellent agreement ^of T with the

percolation prediction of 2.2 suggests ^that three-dimensional percolation ^is a better theory ^for

the critical region than the classical Flory-Stockmayer approach [5, 6] from which

T = 2.5 results.

On the other side, _we found that the classical theory in its extension by Gordon's cascade

theory [7-9] works well for predicting the scattering behavior of the polycyanurate solutions.

Although the assumption of Gaussian chain statistics, which ignores all volume effects, _seems

to be unrealistic, the calculated scattering _{curves are} in excellent agreement with experimental

data. Apparently the fractal dimension D

= 4.0 of _a single ^{cluster is} counter-balanced, if _we average over the broader molecular weight distribution resulting from the Flory-Stockmayer approach.

Thus, it appears possible to use the polycyanurates with their simple, selective chemistry and their well defined broad molecular weight distribution for checking the predictions of both the classical and the three-dimensional percolation theory. ^The present _paper now deals with the critical behavior of this polymeric system ⁱⁿ two ways : by analyzing (I) the

divergence of the weight _average (M~) and the characteristic (M* ) molecular weights and of the radius of gyration (R~), ^{and (iii the} power ^law behavior of the time correlation function

and the divergence of the average relaxation time at the gelation threshold. In that way, ^the

scaling _{exponents were} measured separately by different experiments for the _same polymeric

system ^and can be compared with the theoretical predictions of their values and interrelations.

Experimental section.

The dicyanate used for the synthesis of the polycyanurate polymers, 2,2-bis(4- cyanatophenyl)propane I (dicyanate of Bisphenol A, DCBA) _was synthesized with the help of the cyanbromide method [10]. The resins _were prepared in bulk _at 463 K in _a stirred 100 ml flask which _was heated in _an oil bath. The reaction _was stopped after reaching _a highly viscous state by cooling the flask with liquid nitrogen. The _same procedure _was used for the synthesis

of the copolymers of DCBA with different admixtures of the monophenol 2-phenyl-2-(4- hydroxyphenyl)propane 2 (4-Cumylphenol, CP) (see Fig, I).

For measuring the time correlation function during gelation by online dynamic light scattering a DCBA-prepolymer (M~o

=

420 000 g/mol) at a concentration _c

= 35 g/l was

placed into the measuring cell (2 ml) and 40 mg of the diamine 1,8-diamino-3,6-dioxaoctan

were added. The mixture gelled after _ca, 12 h. A measuring time of 20 min for each _{run was} taken.

The static and dynamic light scattering experiments were made simultaneously with

solutions of the polymers in tetrahydrofuran (THF) at 293 K with _an automatic goniometer

and _a structurator/correlator ALV-3000 by ALV-Langen. The red line (A

=

647 nm) of _an

krypton ion laser Model 2000 by Spectra Physics _was used _as light _source. The solutions _were filtered before measuring through a millipore 5 _~Lm filter. The online measurements during gelation _were done at a scattering angle of 90°.

~i"")

N/

OI

3n NEC-O-R-O-C+N -

~jS~

~ll~Q~/~~~~

,R~ 'R,

/~, ^{° °} /~@

l'~,,~ ~,J~

' n

~~~-CWN ~~~~~

NWC-°~j~~

l 2

Fig, I. Scheme of the polycyclotrimerization ^of dicyanates and structures of the used _monomers DCBA (I) and CP (2).

Gelation behavior static measurements.

Gelation of _a trifunctional randomly branched polymeric _system is known _{as a} typical critical

phenomenon. In both the classical Flory/Stockmayer [5] and the three-dimensional perco- lation theory [I I] the gelpoint of _a covalently crosslinked polymer is defined by the divergence of the second moment of the molar _mass distribution, but, _on the other side, _a different

critical behavior is predicted [I Ii.

Pure DCBA-based polycyanurates _were shown to have _a critical conversion of the cyanate group ^at the gelpoint of 50 fb for the reaction in bulk [3], which exactly coincides with the

prediction of the classical theory [12, 13]. Unfortunately, the required _accuracy, which is needed _to calculate critical exponents, can not be achieved by spectroscopy as pointed out in

our previous _paper [Ii. To _overcome these difficulties the polymeric system was slightly

modified by admixing the monofunctional phenolic analogon CP _to the difunctional cyanate DCBA. In this way, the critical conversion of the OCN-group can be replaced by the critical initial fraction of hydroxyl _groups, since the mixture forms _a gel at full conversion below this critical value only. On the other side, completely soluble polymers with high _mean molecular

weights can be synthesized, if _{one uses} mixtures containing _more monofunctional units than the critical fraction. The determination of the distance to the critical value is _{now a} weighing

process, which _can be done with _a higher _accuracy than the conversion measurement by

infrared spectroscopy.

The _mean molecular dimensions of the polymers _are only negligibly influenced by the

admixtures of the monofunctional _monomers. This is demonstrated in figure 2 where the radii of gyration of pure polycyanurates and of the copolymers _are plotted _versus the weight

average molecular weight. All polymers follow the _same scaling ^relation. Furthermore, from measurements with the help of size exclusion chromatography coupled with light scattering

iO~

E

~ iO~

io~

iO~ iO~ iO~ iO~ iO~

h4w

Fig. 2. Radius of gyration R~ versus molecular weight M~ for pure DCBA- (m) and DCBA/CP- (D) polycyanurates. The regression line follows relation (1).

jQ8

bflmw iQ~

jQ6

~/W

jQ5

lo~~ lo~~

c

Fig. 3. Increase with decreasing relative distance _e to the gelation threshold of the _mean molecular

weight M~ (m) and the molecular weight M~~ (D) corresponding to the maximum of the light scattering

curve of the GPC/LALLS chromatogram. The regression lines follow the relations (3) and (4).

the _same power law for the molecular weight distribution is found _as shown by figure 3 in _our

previous _paper [I ].

The divergence ^of the weight _average molecular weight at the gelation threshold is shown in

figure 3. Here, _s

= (X-X~)/X~ is the relative distance of the normalized initial molar fraction of hydroxyl _groups X

= [OH ]o/([OCN _Jo + [OH _Jo) to the critical value X~. The

optimum values of X~ and _y, which _are highly correlated, are calculated by _an iteration procedure minimizing the residual _mean deviation of the experimental points to the

regression line _as illustrated in figure 4. In that way, a scaling law

M~

~ s~ ^? with _y

= 1.7 _± 0.04 (3)

was obtained, where the exponent y is in very good agreement ^{with the} three-dimensional

percolation prediction but remarkably differs from the Flory/Stockmayer value of 1.

-1 o-lo

I O.QB

i _',,,, ~

,, j

"", _{,,'~ O.OfS} )

~ _~ ',,, _,,,

",,, ,/" O.04 _c

E ',,,~ _,,' [

)

O.02

~

-3 O.OO

24.O 24.1 24.2 24.3

Xc

Fig. 4.- Dependence on the choice of the critical initial molar fraction of hydroxyl _groups X~ of the slope _y defined by relation (3) (-) and of the _mean deviation between experimental and

calculated points (----). The range of the optimum value of _y corresponds to the range of the minimum of the deviation between the dotted lines.

Furthermore, the divergence of the molecular weight M~~ corresponding to the maximum of the light scattering _curve of the GPC/LALLS chromatogram is shown in figure ^{3. This}

value _was shown by Schosseler _et al. [14] _to be proportional to the cutoff molecular weight M*, the divergence of which defines _an exponent _«. The experimentally found relation _:

M~~~

~ s

~'" with _«

=

0.52 _± 0,12 (4)

lies _near the predictions of the classical («

=

0.5 ) as well _as the three-dimensional percolation theory («

= 0.46) but its _error level is higher than the difference between both theoretical values _so that _a clear distinction is _not possible with the help of _« alone.

On the other side, the exponent T, which describes the width of the molecular weight

distribution (2), should be directly calculated from the M~~~ dependence of M~ _as pointed out

by Schosseler _et al. [14]. The least squares fit through the data shown in figure 5 gives the relation _:

M~

~

M(£~ with _T

=

2,18 _± 0.05 (5)

This value agrees very well with that calculated directly from the slope of the molecular

weight distribution.

The experimental data which show the divergence of the _mean cluster size R~ ^{at the} gelation

threshold _are plotted in figure 6 together with the regression line

R~ ^{s~ ~ with} v =

0.9 _± 0.08. (6)

The exponent v also coincides with the three-dimensional percolation prediction (v

= 0.88) [1Ii and lies significantly higher than the classical value (v ₌ 0.5 ). In that way, the basic relations between the different exponents [I Ii _are fulfilled using our experimental

values _:

p = (T 1)/3

=

0.41 p~~~ =

0.41 _± 0.02 (7)

p = Ma = 0.38 (8)

« =

(T 1)/3 _v

=

0.45 «~~~ =

0.52 _± 0.12 (9)

« = (3 T)/y

=

0.46 (10)

iQ~

~~6

~~5

~~4

iO~ iO~ iO~ iO~

~Amax

Fig. 5. Mean molecular weight M~ _versus molecular weight M~~ corresponding to the maximum of the light scattering _curve of the GPC/LALLS chromatograrn. The regression line follows relation (5).

iQ~

~ ~~2

~~~

io~~ io~~

Fig. 6. Increase with decreasing relative distance _e to the gelation threshold of the radius of gyration R~. ^The regression line follows relation (6).

Gelation behavior dynamic light scattering.

It has been shown previously ^{that for} covalently _as well _as for physically crosslinked networks both the power law behavior of the time autocorrelation function (TCF) of the scattered electric field and the divergence of the characteristic relaxation time _are criteria for the

gelpoint [15-19]. The TCF is obtained from dynamic light scattering [20] and may

approximately be regarded as a Laplace transform of the relaxation spectrum in the observed system. ^{For solutions of} polydisperse macromolecules having a broad distribution of

relaxation times the corresponding TCF _can be written _{as a} superposition of exponential decays

gi (t)

= j~ ^{h( Tj} exp (- t/Tj dT (I I)

o

where h(T ) ^is a distribution function. Such _a superposition can often be well approximated by

a so-called stretched exponential function [21, 22]

hi (ti

= a exp i- (t/bi~i ₊ d (121

Here, a is the correlation strength, b is related _to the _mean relaxation time (T) of the spectrum through

jTj

= (b/cj r(i/cj (13j

where r is the _common gamma function, _c contains inforrnations _on both the _mass

distribution of the polymer and the hydrodynamic interaction, and d describes _a possible non- zero baseline.

In _a previous paper [19] _we introduced _a hybrid of _a stretched exponential and _a power law

gi(t)

= hi(t) h~(t) h~(t)

= (tie)~f~~~~ C (t) ₌ I exp(- t/g) (14)

to describe the TCF in the whole range from the pregel through the gelpoint into the gel state with just _one single _type of function. Here, hj(t) is _a stretched exponential defined by equation (12) which approximates _on the _one side the whole TCF far away from the gelation

threshold and _on the other side the initial fast decay, if the power law works in the vicinity of the gelpoint. The function h~(t) is _a power law with _a simple cutoff C (t) of the exponent ^that guarantees a continuous and smooth transition from the power law behavior to the stretched

exponential decay which is valid at short times. In the _same way, the parameter

e rescales the time _so that both parts hi (t) and h~(t) of the hybrid form _a uniform function.

A selection of time correlation functions measured during the chemical gelation of the

polycyanurate in THF is shown in figure 7. From this double logarithmic plot it is easy ^to see

qualitatively that the TCF goes from _a completely decaying behavior during the pregel state

through a power law _{over a} wide range of times _at the gelpoint into _{a non} decaying form with

a non zero plateau in the gel state. It should be noted that the _same kind of _{curves was} obtained by Lang et al. [18, 19] during the thermoreversible gelation of _a polysaccharide.

iO°

gel

a

increasing j

~ ~o-i ^{reaction time *~}

~ ^°

°a I

~ %

o m

pregel _o jo

&

$(

~~-2 ^°

iO~~iO~~iQ~~iO~~iO~~iO~~lO~~ iO° lO~ iO~

t / _s

Fig. 7. Dependence on the reaction time of time correlation functions gj(t) of _a polycyclotrimerizing polycyanurate in THF. The measured _curves correspond to reaction times of 14 and 627 _mn (D), 673 _mn (m), 721 and 821 _mn (+). The function gj (t) shows _a power law behavior _over four decades in time with

a slope of f ₌ 0.34 in the vicinity of the gelpoint (m).

With the help of the model function (14) all measured _curves could be well fitted and the

development with increasing ^{reaction time of the} optimum _parameters is shown in the

figures 8a-c. As _{one can see,} all parameters ^show a characteristic _course, what again

demonstrates that they are not arbitrary fitting parameters ^{but have} a physical or

mathematical significance connected with the kind of curves under consideration.

a, d b, e, Q

i-o io-~

iQ-2 Q ,l'~

',, ---.---'~----.-~-fi~W(.~~W~.

°.~ ~"., iQ~3

Q ~

a "',,

b ' "

',.

"'~'~"~,~ lO~~ ')

~'~

~ i

lo' ,° h

Q4 Q--a"~"~' _lO~~ '~

,°"

d ,,'°'

lo-?

,Q" /

02

lO~~ e

a

OO i

O 300 SOO 900 O 300 SOO 900

reaction time / _min reaction time / _min

a) b)

C >~

i o

off

OA

~

/,,(

, ',

o ~ ^/ a

~ ,"~ "~~

"D o-o

o 300 aoo goo

reaction time / min

C)

Fig. 8. a-c) Dependence _on reaction time of the optimum ^fit parameters according to equations (12) and (14).

The correlation strength a decreases slowly during ^{the initial} part ^{of the reaction and} reaches _a significantly lower value in the vicinity of the gelpoint. The opposite behavior is, of

course, obtained for d, ^which represents ^the contribution of the power law and, after the

gelpoint, the gel fraction respectively. The _non,zero value of d _at the beginning of the reaction is due to the experimental noise of the baseline.

The characteristic time b, from which via equation (13) the _mean relaxation time of the fast motions _can be calculated, as well _as the appropriate exponent c remain nearly constant

during the whole range of reaction times. This behavior indicates that the translational

diffusion with strong hydrodynamic interaction (ZIMM modes with _an expected exponent

c ₌ 2/3) [23] of the small particles inside the solution is not significantly influenced by the

formation of the macroscopic gel during the _crossover from the semidilute regime and

M~

=

420 000 into the gel state including a large fraction of sol.

The increase of the _mean relaxation time of the whole system during the reaction progress up ^to the gel point is reflected by the increasing effect of the power law _term

h~(t) what is manifested in _an increase of d, _e and f. Especially, the sharp increase of the exponent f and the rescaling time _e in the vicinity of the gelpoint is _a quantitative measure for the power law behavior _{over some} decades in relaxation time being a characteristic of the gel point. This fact is substantiated by the behavior in the gel region, where the power law

fraction _now decreases with advancing reaction time but the gel fraction increases,

It must be noted that the slope of the linear part ^{in the double} logarithmic plot of the TCF at the gel point f= 0.34± 0.02 is the _{same as} found for _a thermoreversibly gelling polysaccharide [19] and lies close to the value reported on semidilute solutions of

polyurethane clusters [16] (f= 0.23-0.38). However, it is two times higher than the

experimental value for covalent silicon gels [17] ~f= 0.27 for the homodyne ^TCF g(t)~ [gj(t)]~). Unfortunately, the long time tail of the correlation function _{was not}

sufficiently accessible _on the experimental time scale. Therefore, _an analysis of the divergence

of the slowest characteristic time _{was not} possible with acceptable _accuracy.

The scaling behavior of the mean relaxation time (T) of the spectra at the gelation thresh- old is shown in figure 9. Here, (T) _was calculated by an integration of the TCF _over the

whole experimental time window

j~j

=

~~ _{gj(t) dt (15)}

and _s

= (t~ t)/t~ _represents the relative distance _to the critical reaction time t~. ^From a

regression analysis done in the _{same manner as} described for the calculation of X~ and y (see Fig. 4) the relation

(T) ^{s~~ with} #i = 1.9 _± 0.2 (16)

was obtained. The _same exponent was found by Martin et al. [17] for covalent silicon gels but through an integration of the homodyne TCF.

io'

m iO°

) iO~'

v <~>

lO~~

<~>f

u

~Q-3 m

io~~ io~' io°

c

Fig. 9. Mean relaxation time (T ) of the whole function (m) and (T_)~ of the fast motions (c) _versus relative distance _e to the critical reaction time. The regression line for the divergence of

(T) ^{follows relation} (16).