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Size and mass determination of clusters obtained by polycondensation near the gelation threshold
M. Adam, M. Delsanti, J.P. Munch, Denys Durand
To cite this version:
M. Adam, M. Delsanti, J.P. Munch, Denys Durand. Size and mass determination of clusters obtained by polycondensation near the gelation threshold. Journal de Physique, 1987, 48 (10), pp.1809-1818.
�10.1051/jphys:0198700480100180900�. �jpa-00210622�
1809
Size and mass determination of clusters obtained by polycondensation
near the gelation threshold
M. Adam (1), M. Delsanti (1), J. P. Munch (+,1) and D. Durand (2)
(1) Service de Physique du Solide et de Résonance Magnétique, CEN-Saclay, 91191 Gif sur Yvette Cedex, France
(2) Laboratoire de Chimie et Physico-Chimie Macromoléculaire, Unité associée au CNRS, Université du Maine, route de Laval, 72017 Le Mans Cedex, France
(Reçu le 8 avril 1987, révisé le 18 juin 1987, accepté le 19 juin 1987)
Résumé. 2014 Au voisinage et au-dessous du point de gel, nous étudions le processus de gélification par
polycondensation conduisant à des amas de polyuréthane self similaires. Les expériences de diffusion de lumière sont faites en solutions diluées. L’interaction entre amas est une fonction de l’écart au seuil de
gélification, 03B5. Ceci est une observation directe du gonflement de ces amas. Il est alors évident que la valeur mesurée de l’exposant reliant la taille des plus grands amas à l’écart au seuil de gélification 03B5 est plus grande
que celle prévue par le modèle de percolation. Par contre, l’exposant 03B3 reliant la masse moyenne en poids des
amas à 03B5, insensible au gonflement, est trouvé égal à 03B3 = 1,71 ~ 0,06, valeur en bon accord avec la théorie de la percolation. Gélification par polycondensation et percolation appartiennent à la même classe d’universalité : elles partagent 2 exposants de même valeur 03C4 et y.
Abstract. 2014 The gelification process by polycondensation leading to self similar clusters of polyurethane is studied, near and below the gelation threshold. In dilute solutions, where light scattering experiments are performed, the interaction between clusters is a function of 03B5, the distance to the gelation threshold. This is a
direct evidence of the swelling of the clusters. Thus, the experimental value of the exponent which links the size of the largest clusters to the distance from the gel point 03B5 is larger than the value predicted by the percolation theory. However, the exponent 03B3, which links the weight average molecular weight to 03B5, is
determined and found to be equal to 1.71 ~ 0.06, in agreement with percolation theory. The gelification
process by polycondensation and percolation belong to the same class of universality : they share two exponent values 03C4 and 03B3.
J. Physique 48 (1987) 1809-1818 OCTOBRE 1987,
Classification
Physics Abstracts
61.40 - 82.35 - 05.20
Half a century ago a mean field theory of the gelation threshold [1, 2] was developed. This theory,
well adapted to describe the chemical process (as for example the location of the gel point) was verified experimentally [3]. About ten years ago an analogy
was done [4, 5] between gelation and percolation,
the latter being a critical phenomenon of connec- tivity. This analogy was followed by Monte Carlo
simulation [6] and by few experiments. Here we are
interested in the formation of gel through chemical
reaction.
We show experimentally that the gelation process,
near and below the gelation threshold, is well
described by the critical theory of percolation.
1. Theoretical background.
Percolation is a critical phenomenon of connectivity,
all the characteristic quantities of this critical phenomenon are expressed as power laws of the distance to the critical point. Moreover one has to
know the variation of only two of these quantities,
and thus the corresponding exponent values, in order to know the behaviour of all the other characteristic quantities.
First of all we will describe the percolation model
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100180900
and quote the behaviour of the measurable quantities
in the experiments reported here.
In the percolation model a certain number of
bonds are thrown on an N site, finite three-dimen- sional (d = 3) lattice. Two bonds which are nearest neighbours are considered connected. A cluster is an
ensemble of connected bonds. p is the ratio of the
, total number of bonds thrown on to N. At the
threshold, characterized by pc, there exist a giant
cluster. Near and below the threshold the system is completely described by the mass distribution [7] :
where n dM is the number of clusters of mass M to M + dM (Mass corresponds to number of bonds),
M * is the biggest mass present in the system and f is
a cut-off function. The calculated value of the T
exponent is 2.20 [7]. The variation of M* with the distance to the threshold
with
The size and the mass of the ith cluster are linked by Ri ~ MP with p = 0.40. The fractal dimension is defined by
The correlation length ç:
is proportional to the size of the biggest cluster, it
varies as :
One can define a mean molecular weight by
The variation of Mw with e is given by :
Above the percolation threshold there exist two
phases, the first one is identical to that described above, the second is a multiconnected cluster having
the size of the lattice. The probability for a bond to belong to this cluster is given by
with
The exponents values given above (listed in Ref.
[7]) are obtained by Monte Carlo simulations at a
space dimensionality d of 3, their values are function of d (for example y = 2.389 for d = 2, 1.74 for
d = 3, 1 for d = 6). At d = 6, the exponents values
are identical to those deduced from the Flory-Stock- mayer theory [8], because the mean field theory is
valid for this dimensionality of space.
In this study we are interested in the gelation
process near the gelation threshold. Here p is the degree of advancement of the chemical reaction and pc its value at the gelation threshold.
The mean molecular weight MW and the correla- tion length § are measured by light scattering inten- sity experiments. In order to perform these experi-
ments one has to dilute the sample in a solvent in
order to create a contrast and to disimbricate the clusters. This dilution has two effects, first to swell
the clusters and thus decrease the fractal dimension D [9] (Mi ~ Ri D), second to remove the shielding of
the interaction between the clusters. In dilute solu-
tion, the interaction parameter B is calculated using
the expression of Yamakawa [10]
where JY’a is the Avogadro number and Vij is the
excluded covolume between the ith and the jth
clusters. The interactions between two monomers
belonging to the largest cluster are screened by all
the smallest clusters, the covolume Vij’ for Mi > Mj
is [11]
where Ri is the radius of the swollen cluster.
Inserting the laws obtained in the framework of
percolation model (relations (1), (7) and (3)) in
relations (9) and (10), for ni and MW we obtain for
the variation of B as a function of the distance to the
gelation threshold E [12]
with
One can be easily convinced that, as in a linear monodisperse polymer system of mass M and size R,
the inverse of B (-MIR 3) represents the internal concentration of monomers averaged on the cluster distribution. As in linear monodisperse systems, B is proportional to the intrinsic viscosity (Ref. [8]
p. 145).
1811
In order to study the gelation process we chose a
system in which the chemical reaction is a polycon-
densation.
2. Experimental procedure.
2.1 THE CHEMICAL SYSTEM AND LOCATION OF THE GEL POINT. - The chemical system was fully de-
scribed in reference [13], here we only recall the main features.
Polyurethane is prepared, without solvent, by
condensation of OH groups belonging to trifunc-
tional unit (poly(oxypropylene)-triol with a molecu-
lar weight of 700) and NCO groups belonging to
difunctional unit (hexamethylene diisocyanate with
molecular weight of 168).
One can define the stoichiometric ratio as r = [NCO] [OH] where [NCO] and OH are the con-
centrations of the NCO and OH groups initially present. r is determined by weighing, its precision is
5 x 10- 5. If r 1, at complete reaction, r is pro-
portional to the number of the OH groups which have reacted. The verification of the absence of the NCO groups at complete reaction was given else-
where [13]. The sample preparation is reproducible
because samples prepared in a time interval of 6 months, using the same batch of dried monomers,
give coherent results (see for example Fig. 4). A comparison between the definitions of r and p shows
that these two quantities are proportional. The main problem is to locate the r value corresponding to
rc, the gelation threshold. For r 0.5593 the system
can be dissolved in dioxan (a solvent), for r ;::= 0.56
the system cannot be dissolved
In order to determine the exponent values which link the measured quantities to E we set
The different solutions are prepared in the follow- ing way : a first dilution in stabilized 1-4 Dioxan
(Merck) is prepared at a concentration C
(0.01 C (g/cm3 ) 0.1 ) depending on the r value (lower r higher C). Then, using Millipore filter (Millex - SR 0.5 )JLm), this first solution is filtered into the scattering cell in which the desired amount of filtered solvent is added. The concentration is determined by weighing and converted into g/cm 3
through C (g/cm3) = p C (g/g ) where p, the density
of dioxan, is equal to 1.034 at 20 °C.
2.2 LIGHT SCATTERING EXPERIMENTS. - The spect-
rometer is a home built apparatus, described else-
where [14], whose specifications are reported in the appendix. An argon ion laser with an incident power of about 200 mW is used at A = 4 880 A. The angle
selections and intensity measurements are controlled and governed by a computer.
The intensity, ns, scattered by the sample at a
diffusion angle 0 is normalized by the incident intensity ni and corrected for the variation of the
scattering volume :
cal alignment of the spectrometer is checked using a
benzene sample. Usually we find that N benzene «(J ) is
a constant having a precision of 0.8 % within the
angle range of 10° to 159° . When the measurements are performed on a polymer sample, diluted in
dioxan, the following relative intensity is calculated
The relative intensity 1 (0, C ) scattered by a polymer
solution is a function of the scattering angle and of
the polymer concentration C.
Let us first consider the case of a monodisperse polymer having a molecular weight M. The relative intensity scattered, extrapolated to zero angle, is given by :
where B is the interaction parameter. The constant
Rayleigh ratio of benzene equal to 38 x 10- 6 cm-1
at A = 4 880 A [16], n the refractive index of dioxan
(1.425 at À = 4 880 A) and an was determined ac
using an Abbe refractometer at A = 5 000 A. In
figure 1 the refractive index increment is plotted as a
function of the concentration. We find
an
= 4.5 x 10-2 (cm3/g) independent of r. There is
ac
Fig. 1. - Increment An of the index of refraction as a
function of the polyurethane concentration C. The sym- bols A, EB, (p correspond to samples having a stoichio-
metric ratio r of 0.548, 0.499 and 0.452, respectively. The straight line corresponds to a refractive index increment
=4.5xl0-’(cm’/g).ac
no measurable difference of an for diisocyanate and
ac
for trialcohol. Using the numerical values listed
above we find :
The intensity scattered, extrapolated to zero concen-
tration, is angle dependent and allows the form factor P (0 ) of the polymer to be determined
As the form factor, P (0 ), is an unknown function (except for a linear Gaussian polymer) the following expansion form is used :
q is the transfer vector equal to 4 À 7T’ n sin (0 /2)
(278 q -1 (Å) - 2 958, in the range of 0 investi-
gated 10° 0 159°). R is the radius of gyration of
an isolated polymer. The ratio a = L /R depends on
the conformation of the polymer: for a linear
. polymer in a theta solvent or in a good solvent a is equal to 1 within 2 % [17].
Actually experiments are performed at finite con-
centration and with a highly polydisperse system.
The quantities measured are mean values :
- the molecular weight M is in fact a mean
weight average molecular weight Mw
where ni is the number of polymers having a
molecular weight Mi ;
- the characteristic sizes R and L are z average
mean values :
and
At a given concentration, the relative intensity is
fitted to :
or
with
One should note that the intermolecular interac- tion modifies the form factor. By making this kind of treatment we do not impose any law for this modification.
In order to check the analysis and the qRZ (C )
range of validity of formulae (17) and (18) we use a sample having a mean radius of gyration of about
700 A.
- With the first approximation (Eq. (17)) we
find (see Fig. 2a) :
for 0.2 qRz 1,
and
- with the second approximation (Eq. (18)) we
find (see Fig. 2b) :
for 0.2 S qRZ 2,
and
and
with a sample having a mean radius of gyration of
1 000 A we find that the second approximation (Eq.
(18)) can be extended to qRz ~ 3.
Within experimental precision, both approxima-
tions give identical results. The second is preferen- tially used because it allows the whole dynamic of
the experimental form factor to be exploited. The reproducibility of the experimental results obtained
with our spectrometer using the analysis described above, is 4 %, 6 % and 10 % for the z average radius of gyration, the weight average molecular weight
and the interaction parameter, respectively.
By comparison of relation (6) with relation (14),
the mean molecular weight Mw measurable by a
1813
Fig. 2. - Example of q2 dependence of the relative
intensity scattered. In linear, scale is plotted 11I (in arbitrary unit) as a function of sin 20/2. The maximum
qRz value in figures a and b are 0.87 and 1.93, respectively.
The full lines in Figures a and b correspond to the best fits using equations (17) and (18), respectively.
light scattering experiment, we can see that both
quantities are proportional.
By comparison of relation (4) with relation (15),
the mean size Ri measurable by light scattering experiment, we can see that Ri is proportional to ç 2 only if the ith cluster has the same fractal dimension in diluted solution (where Ri is measured)
and in the percolation model.
3. Experimental results.
During the gelation process and before the gelation
threshold the system is fully polydisperse, the
molecular weights extend from that of the monomers
to the molecular weight of the biggest cluster
M*. For a given distance to the gelation threshold
the number n of clusters having a
molecular weight M is given by the mass distribution
equation (1) with an exponent value :
determined experimentally [18] by neutron scattering experiments on a sample elaborated at e ~ 2 x 10- 2
and diluted in deuterated tetrahydrofurane (TDF).
In the reaction bath (undiluted sample) following
relation (3) using T = 2.20 ± 0.04, the fractal dimen- sion Dp (Mi ~ Ri Dp), is found to be equal to
2.5 ± 0.09.
The fractal dimension of swollen clusters was also measured [18] :
The relative intensity of the light scattered by a
diluted sample, extrapolated to zero transfer vector
q, is strongly dependent on concentration (see Fig. 3a) and presents a maximum at a concentration denoted as C. It is found that C is inversely proportional to B, the interaction parameter. The experimental results (Fig. 3b) used for the extrapola-
tion to zero concentration are obtained in a concen-
tration range smaller than C (thus C 2 terms in
relation (13) are negligible).
In figure 4 we plot the variation of the weight
average molecular weight MW as a function of the
distance to the gel point. The full lines in the figure correspond to the law
variation of r c leads to The interaction parameter B is dependent on the
distance to the gel point s (see Fig. 5). We find
A variation of rc of :p 3 x 10-4 leads to an exponent value of 0.59 + 0.02.
The experimental exponent value (Eq. (23)) is in good agreement with that obtained by inserting the
T, y and D experimental values in relation (11).
One has to note that intrinsic viscosity measure-
ments performed on the same samples prepared at
different values of r (2 x 10-2 _ E 0.1) diluted in tetrahydrofurane (THF) lead to
The e independence of the ratio B means that
the fractal dimension D of clusters of polyurethane is
identical in the two solvents used (THF and Dioxan).
These experimental results (on B and [n]) are direct
Fig. 3. - Typical variation of the relative intensity scat-
tered extrapolated to zero q value as a function of the concentration. The intensity I ( 0 - 0 ) (symbol 9) presents
a maximum for C = 11. In figure 3b are reported
C x I - 1 (0 -+ 0 ) (symbol +) and Rz as a function of concentration. Both quantities are linear functions of the concentration for C C (experiments performed with a sample prepared at E = 3.36 x 10- 2) .
observations of the swelling of the clusters because
the two quantities are independent of c in the
unswollen state (see Ref. [8] p. 145 and relation (11)).
Due to this swelling, the exponent v’ linking the
size to the distance to the gel point cannot be directly compared to the percolation exponent value
v = 0.88. Actually, using our experimental results
on the z average radius of gyration (see Fig. 6)
Fig. 4. - Divergence of the mean weight average molecular weight Mw as a function of s the distance to the
gel point The representations are log- log scale and linear scale in the inset. The full lines
correspond to the relation (22). Different symbols corre- spond to sets of preparation in which the same batch of dried monomers was used but the preparation was done in
a 6 months time interval.
Fig. 5. - Variation of the interaction parameter B
(cm3 g-1 ) as a function of the distance to the gel point e in
a log-log scale. The straight line corresponds to the
relation (23).
Fig. 6. - Divergence of the Z average radius of gyration
as a function of the distance to the gel point e =
The representations are log-log scale and linear scale in the inset. The full lines correspond to relation (25) (for the meaning of the symbols see Fig. 4).
1815
extrapolated to zero concentration, we find using
A variation of rc of + 3 x 10-4 leads to a variation of
0.04 of the exponent value.
The linear behaviours, in the log log representa- tions (Figs. 4, 5 and 6) indicate that rc evaluation (0.5596) is correct and its precision is adequate in the
range of e investigated (4 x 10- 3 to 10-1 ).
The ratio of the different mean values of the sizes
Lz (Eq. (16)) and Rz (Eq. (15)) is a constant in the
whole range of e (4 x 10-3-2 x 10-2) where both
quantities can be determined
In relation (1), supposing that f (M/M* ) is a
constant for M M *, and equal to zero for
M > M* we find that:
where a is a parameter dependent on the conforma-
tion of the clusters. Using relation (26) and the
values of T and D (Eq. (20), (21)) we obtain [19]
in agreement with the experimental results (one can
suppose here that a ~ 1 as it is for both a Gaussian
polymer and a swollen polymer).
This experimental result, the ratio of z average
sizes, Lz/ Rz, is a constant independent of E, seems to
indicate that the mass distribution function, once
normalized by the number of clusters having the
mass is in-
dependent of the stoichiometric ratio r. Or, in other words, the clusters obtained by polycondensation for
different stoichiometric ratio r of diisocyanate and
triol are mass self similar.
In figure 7 we report the variation of Mw as a
function of Rz on a log log scale. In this figure are
also reported results obtained on 2 samples on which
the radii of gyration were measured by small angle
neutron scattering experiments (samples diluted in TDF) and molecular weights by light scattering experiments. This allows us to increase the range of
Mw and R, investigated [20]. The straight line corresponds to the law
One can show that this exponent (1.61) corresponds
to (3 - T ) D. This exponent was also directly
Fig. 7. - Relation between mean Z average radius of
gyration Rz and mean weight average molecular weight Mw. The representation is a log-log scale and straight line corresponds to the relation (27). Symbols 0 correspond to samples in which Mw was measured by intensity light scattering and Rz by small angle neutron scattering [20].
measured by small angle neutron scattering experi-
ments on one sample diluted in TDF (E::- 2 x 10-2)
[18], it was found to be D(3 - T ) = 1.59 + 0.05.
The identity of exponent values found using two
different approaches (either one sample
e ~ 2 x 10-2 or different samples 4 x 10-3
S ê S 10-1) and two different techniques (neutron
and light scattering) make us confident in those
experiments. Moreover this indicates that the fractal dimensions D and Dp (via T) do not depend on E.
One can conclude that the geometry and the growth
of clusters are independent of r, for
We have measured two exponents T (2.20 + 0.04 ) and y (1.71 ± 0.06 ) which are in agreement with
Table I. - Comparison between theoretical expo-
nent values and values determined either experimen- tally (a) or deduced inserting the experimental values
into the following relations :
