A Model for Phase Separation During a Thermoset Polymerization

13

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 23, 2015 | http://pubs.acs.org Publication Date: December 5, 1984 | doi: 10.1021/ba-1984-0208.ch013

A Model for Phase Separation During a Thermoset Polymerization R. J. J. WILLIAMS, J. BORRAJO, H . E. ADABBO, and A. J. ROJAS Institute of Materials Science and Technology (INTEMA), University of Mar del Plata—National Research Council, J.B. Justo 4302, (7600) Mar del Plata, Argentina

A model was developed to predict the fraction and composition of the dispersed phase segregated during a thermoset polymerization (e.g., a rubber-modified epoxy system). This model can also be used to predict the mean radius and volumetric concentration of the dispersed particles. The location of equilibrium and spinodal curves is described by a Flory-Huggins equation that includes the thermoset conversion. Equations for nucleation, coalescence, and growing rates were derived and analyzed. The morphology development is assumed to be arrested by gelation of the thermoset matrix. The dispersed-phase fraction and the concentration of dispersed particles decrease with an increase in the curing temperature; however, the mean radius goes through a maximum. Accelerating the polymerization by adding a catalyst has the same effect as a temperature increase. Conditions leading to spinodal demixing are discussed.

D I S P E R S E D R U B B E R Y P H A S E can improve the toughness and impact properties of cured thermosets. A typical example is the use of low levels of r u b b e r copolymers of b u t a d i e n e - a c r y l o n i t r i l e i n epoxy resins. Initially the system is homogeneous; but, at a certain ther moset conversion, rubber-rich domains begin to be segregated from the matrix. T h e morphology development continues until the ther moset matrix reaches a conversion close to the gel point ( J ) , or vit rifies (if the glass transition takes place before gelation). The mechanical properties of the resulting specimen are strongly dependent on the morphology developed during the phase-separa tion process. Both formulation and curing conditions affect the re sulting morphology (1-3). Therefore, prediction of convenient for mulations and/or curing conditions to obtain desired morphologies is important.

X J L

0065-2393/84/0208-0195/$06.00/0 © 1984 American Chemical Society

In Rubber-Modified Thermoset Resins; Riew, C. Keith, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1984.


196

RUBBER-MODIFIED THERMOSET RESINS

A model for phase separation during a thermoset polymerization is analyzed i n this chapter. A l t h o u g h the model may be adapted to describe different systems, particular attention is given to rubbermodified epoxies. Moreover, for the sake of illustration purposes most of the model parameters are taken from a formulation used by M a n zione et al. (2). This system was a low molecular weight bisphenol A type epoxy resin that was modified with a carboxyl-terminated polybutadiene-acrylonitrile ( C T B N ) w i t h piperidine as the catalyst. F o r this model, w e assume that the rubber is an inert component dissolved i n a monodispersed epoxy resin that homopolymerizes i n the presence of piperidine. Rubber-containing diepoxide resulting from the capping of carboxyl end groups of the C T B N with the epoxy resin is assumed to have no influence on kinetic and statistical pa rameters of the epoxy homopolymerization. Thermoset

Polymerization

The curing rate is assumed to follow a second-order kinetics, dpldt = A (1 - p ) exp(-E/RT)

(1)

2

where p is the thermoset conversion. T h e specific rate constant (A) is 5 x 10 s and E/R = 10 K ; these values are consistent w i t h reported data o n time to gelation (2). B y assuming that the usual s i m p l i f y i n g hypotheses are v a l i d (equal reactivity, no substitution effects, and no intramolecular reac tions i n finite species), w e can characterize the homopolymerization of a monomer w i t h four reactive sites p e r mole (functionality = 4) by the following statistical parameters (4-6): 7

- 1

4

Pgel = 1/3

(2)

M ^ / M ^ = 1/(1 - 2p) = (1 + p)/(l - 3p)

M /M Wp

(3)

Wo

(4)

M and M are the number and weight average molecular weights, respectively. W e have also assumed that, i n this range of curing temperatures, the resin gels before vitrifying; the morphology development w i l l thus be arrested at p . The thermoset viscosity, T), may be written as a function of M , because both parameters become infinite at the gel conversion, n

w

g e l

w

Vn

0

= (M /M / Wp

(5)

w

The following particular functionality w i l l be used for illustration pur poses: ri(kg/m • s) = 2.24 x 1 0 " [(1 + p)/(l - 3p)] exp(Er,/RT) 6

2


(6)

13.

Thermoset

WILLIAMS ET AL.

197

Polymerization

where ET|/R = 4 X 1 0 K . This equation gives viscosity values close to reported experimental data at p = 0 (2). 3

Thermodynamics


The initial system is regarded as a solution of rubber (component 2) in an epoxy solvent (component 1) w i t h a free energy per unit volume (AGy) described b y the F l o r y - H u g g i n s equation. &Gy

= (RT1V ) [(1 - c|>) l n ( l - ) + 2

X

2

( V * ) ln + x 4> (1 2

(7)

)]

2

2

where V is the molar volume of the epoxy resin, ^ is the volume fraction of component i, z is the ratio of molar volumes of both com ponents (V /Vi), and x is the F l o r y - H u g g i n s interaction parameter per mole of solvent (component 1). The parameters V , z, and x vary during the thermoset poly merization. T h e molar volume of epoxy resin changes according to X

2

x

Vi = V

l o

(M„ /M„ ) = V / ( l - 2 p ) p

o

(8)

l o

T h e n , the ratio of molar volumes becomes z = V/V, where z

0

=

2p)

= z0 ( 1 -

(9)

V /V . 2

1Q

The F l o r y - H u g g i n s interaction parameter per mole of solvent may be written as * = *o W

0

= xj(l

2p)

-

(10)

Therefore, b y combining Equations 7 - 1 0 , we obtain HG™ = (Rr/V ){[(l - 2p)(l - ) l n ( l - c[>)] + [ ( c M ^ i n c y + [x (i - f 3

2

c

2c

o

2

2c


13.

201

Thermoset Polymerization

WILLIAMS ET AL.

Z =10 0

T = 373 K 0.2

-


\

0.1

\ s p .

Eq. \

I

l

i

l

i

I

Figure 4. Location of equilibrium (Eq.) and spinodal (Sp.) curves in a rubber fraction vs. conversion diagram. (i.e., the metastable region, or supersaturation, begins). T h e extent of variation depends upon the relationship between the intrinsic rate of phase separation and the polymerization rate. T h e significant de parture of 4> from f (Figure 5) arises from a relatively high poly merization rate. T h e opposite situation w o u l d produce a § value very close to the e q u i l i b r i u m value during the entire phase-separa tion process. B y definition, the intersections of the tangent at and both 2c

2 c

2c


202

RUBBER-MODIFIED THERMOSET RESINS

ordinates represent the partial free energy per unit volume for the pure components i n the continuous phase. The tangent line also gives the virtual free energy per unit volume for any solution with an arbitrary composition, (J> , segregated from the solution with a com position 2 . The continuous curve, A G y vs. , gives the actual free energy value per unit volume for any solution. Therefore, the free energy change involved i n the separation of a dispersed phase of any arbitrary composition, A G ^ , is given by the distance between the tangent line at and the free energy function A G y . F o r example, if the dispersed priase has the composition (J> , the free energy per unit volume w i l l decrease by an amount A G , as is depicted i n F i g ure 5. A l l the values possible for A G are shown i n Figure 6. F o r c|> = , the maximum possible decrease i n free energy is obtained. Thus, °° and D —> 0. g e l

0

g e l

Concentration

of Dispersed-Phase

Particles

The concentration of dispersed-phase particles per unit volume, P, increases w i t h nucleation b u t decreases through coalescence. This last process is proportional to the square concentration of particles and inversely proportional to viscosity (11). Therefore, the rate of variation of the concentration of particles per unit volume may be written as dP/dt = dN/dt - (4/3)(kT/t\) P

(20)

2

Although the numerical factor used i n the coalescence term may not be exact (11), a correct order of magnitude is predicted. Because time and reaction extent are related through polymer ization kinetics, E q u a t i o n 20 may be rewritten by using p as the independent variable dP/dp = dN/dp - (4/3)(kT/Ai)) P [exp(E/RT)/(l - p) ] 2

Because T| —» oo at p = p Particle

g e l

2

(21)

, the coalescence is arrested at gelation.

Growth

W h e n the system evolves through the metastable region, particle growth occurs because of the driving force ( - 4>2)> which tries to 2


13.

205


WILLIAMS ET AL.

restore the system to e q u i l i b r i u m conditions. The growth rate ( G . R . ) , defined as the increase i n volume fraction per unit time, may be assumed to be proportional to the interfacial area per unit volume and the driving force G.R. =

4TTR P ( $ 2

-

2c

cf>!)

(22)

where k^ is a mass transfer coefficient and R is the mean radius of dispersea-phase particles (4TTR P is the surface area per unit volume). T h e mass transfer coefficient for a sphere i n a stagnant m e d i u m , expressed i n terms of a volumetric fraction driving force, is given by


2

(12)

*• = D/R Because D = 0 at p Mean Particle

g e l

(23)

, the growth rate is arrested at gelation.

Radius and Volume Fraction

of Dispersed

Phase

The mean particle radius is given by R = (3 V /4nP)-3

(24)

D

where V is the volume fraction of the dispersed phase at any p. The change i n this fraction w i t h p is given by D

dV^dp

= (4ir/3)(r ) (dN/dp) + ^ Cp

3

4TTR

2

F

(C|>

2C

-

^(dt/dp)

(25)

The amount of dispersed phase increases with nucleation and particle growth. The composition of the new phase segregated at any p is
0

=

Vd

$ 2

D

+

(1

-

V ) D

, suggests that most of the phase segregation takes f

0

11

3

1

_ 1


_ 4

2

2

2d

Figure 8. Evolution of the rubber fraction volume in both phases as a function of the thermoset conversion (low curing temperature).



13.

WILLIAMS ET AL.

Thermoset

207

Polymerization

place w e l l before gelation, i n agreement w i t h experimental results (I). A t high curing temperatures (T = 423 K ) , the instantaneous compositions are different from the e q u i l i b r i u m values, a n d the system has advanced into the metastable region. N o w , phase sepa ration does not restore the system to the e q u i l i b r i u m condition be cause of the high polymerization rate. However, the spinodal curve is not attained because of thermodynamic restrictions (see Figure 2). Significantly, $ remains very close to the initial value, . There fore, the volume fraction of the dispersed phase must be negligibly small. Results show that producing spinodal demixing i n a rubber-mod ified epoxy is not easy. F r o m Figures 2, 4, 8, and 9, the necessary conditions seem to be high initial rubber content ( close to the critical value), l o w c u r i n g temperatures (thermodynamic r e q u i r e ment), and high reaction rates (kinetic condition). 2 c

2o

2o



208

R U B B E R - M O D I F I E D T H E R M O S E T RESINS

F i g u r e 10 shows the evolution of the different parameters that characterize the morphology for an intermediate curing temperature. The mean radius of particles at p = p is R = 1.5 jxm, a value attained w e l l before gelation. This value lies i n the range of reported experimental results (2, 7). The volume fraction of the dispersed phase reaches a final value, V = 0.08. This fraction includes slightly more than half of the rubber initially added to the formulation. O n e possibility of increasing this amount is to operate with a thermoset that gels at a high conversion (i.e., and e p o x y - d i a m i n e system for w h i c h p i is close to 0.58). The influence of the curing temperature on the resulting mor phology at p is shown i n Figure 11. The concentration of dispersed particles decreases continuously with temperature because the po lymerization rate increases more rapidly than the nucleation rate (in Equation 19, E > + E + A G ) . The coalescence rate d i d not seem to be relevant w h e n compared to the nucleation rate, and E and A G were very small values. g e l

D

ge

g e l

F

C

F

C



13.

WILLIAMS ET AL.

209


Figure 11. Final values of different parameters that characterize the morphology vs. curing temperature. The volume fraction of dispersed phase, V , is almost constant at low temperatures, because it is l i m i t e d only b y the binodal curve and the gelation conversion (thermodynamic restriction). However, at high temperatures, V drops abruptly because of the high poly merization rate (kinetic restriction). A s a consequence, the mean ra dius, R, w h i c h depends on the V /P ratio, goes through a maximum as the temperature is increased. This result agrees with experimental observations (2). A l l these trends depend on the fact that E > E^. A thermoset for w h i c h E^ > E w i l l show opposite effects when the curing temperature is increased. A n increase i n the polymerization rate at constant temperature (i.e., by adding a catalyst) has the same effect as a temperature i n crease (Figure 12), because both parameters increase the polymer ization rate i n relation to the phase-separation rate. Finally, increasing the initial rubber amount i n the formulation leads to a considerable increase i n the volume fraction of dispersed phase and concentration of dispersed particles (Figure 13). This obD

D

D



servation is i n agreement w i t h experimental results (7). However, the average rubber concentration of the dispersed phase, 4> , drops significantly. B o t h effects occur because the binodal is reached at lower conversions w h e n is increased (Figure 4). This significant change i n the composition of dispersed domains may have a bearing on the toughening behavior. 2d

2

Summary W e have modeled the process of phase separation during a thermoset polymerization by using simple thermodynamic, kinetic, and statis tical arguments, as w e l l as related constitutive equations. The model may be adapted to different thermoset formulations and operating conditions. F o r rubber-modified epoxies cured under isothermal conditions, the m o d e l predicts the following trends: • Phase separation probably proceeds through a classic n u cleation-growth mechanism rather than through spinodal demixing. Spinodal demixing would require the use of a formu lation w i t h a high initial rubber amount, low curing tempera tures, and high reaction rates. • The influence of the curing temperature on phase separa tion is strongly dependent on the difference of chemical and



WILLIAMS E T AL.

211


Figure 13. Final values of different parameters that characterize the morphology as a function of the initial rubber concentration. viscous activation energies, (E - E^j. If this difference is high, as is the usual situation, a temperature increase w i l l cause a considerable decrease i n the concentration of dispersed-phase particles, a maximum i n the mean radius, and a significant drop i n the volume fraction of the dispersed phase at high tempera tures. These effects are exactly the same when the polymeriza tion rate is increased at constant temperature. In both situa tions, the explanation relies on the increase of the polymeriza tion rate w i t h respect to the phase-separation rate. • Increasing the initial rubber amount, , leads to a con siderable increase i n the volume fraction and concentration of dispersed-phase particles. However, the rubber concentration in the segregated domains drops significantly. If is greater than a critical value, an inversion i n the nature of the segregated phase w i l l result. M o s t of these trends agree w i t h experimental observations (I, 2, 7, 9). 2

2o


212

R U B B E R - M O D I F I E D T H E R M O S E T RESINS

List of

Symbols

Specific rate constant for thermoset polymerization Diffusion coefficient of rubber i n the thermoset Activation energy of polymerization Activation energy associated w i t h composition fluctuations Activation energy of viscosity Composition fluctuations per unit volume Proportionality constant Adjustable parameter Particle growth rate F r e e energy change i n the nucleation process F r e e energy barrier for the formation of a nucleus with critical radius AGjv F r e e energy change associated w i t h phase separation A G y F r e e energy per unit volume k Boltzmann constant k^ Mass transfer coefficient M N u m b e r average molecular weight M W e i g h t average molecular weight N N u c l e i concentration per unit volume p Thermoset conversion (reaction extent) P Concentration of dispersed-phase particles per unit volume r Nucleus radius r C r i t i c a l radius of nucleus R Gas constant R M e a n radius of dispersed-phase particles V M o l a r volume V Volume fraction of dispersed phase x F l o r y - H u g g i n s interaction parameter per mole of thermoset z Ratio of molar volumes of rubber and thermoset

A D E E E^ F F F' G.R. AG AG F


0

0

C

n

w

c

D

G r e e k Letters T| cr

A Model for Phase Separation During a Thermoset Polymerization

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