J. Phys. Chem. B 1999, 103, 3997-4005
3997
Linear Chain and Network Polymerization during Pressure Upstep and Downstep by Real-Time Dielectrometry D. A. Wasylyshyn and G. P. Johari* Department of Materials Science and Engineering, McMaster UniVersity, Hamilton, Ontario L8S 4L7, Canada ReceiVed: July 22, 1998; In Final Form: January 28, 1999
To gain insight into the physical and chemical effects during the linear chain and network formation, dielectric properties of four diepoxide-amine liquid mixtures have been studied during their polymerization at pressure raised from 1 bar to 200 bar in an upstep, and after a predetermined period lowered to 1 bar in a downstep manner, many times during the course of polymerization. For comparison, dielectric properties were also studied when the samples were maintained at a fixed 1 bar pressure or at 200 bar pressure. Although pressure is expected to decrease the polymerization rate for all conditions, the decrease was observed only when polymerization became diffusion-controlled. In the early stages, effects other than the viscosity’s increase on compression dominate the dielectric behavior. An analysis by mathematical simulation shows that both physical and chemical effects of pressure steps are significant, but their relative magnitudes vary during the course of polymerization. A pressure upstep also increased the sample’s temperature, and a downstep decreased it, after which the temperature reached the equilibrium value asymptotically. This effect was also evident in the time dependence of the dielectric properties and was found to be consistent with the effects of temperature observed before. The study demonstrates the validity of the various concepts used for the pressure and temperature effects on the negative feedback effects between molecular diffusion and chemical reaction during a macromolecule’s growth.
Introduction Chemical reactions during the polymerization of a liquid are intimately related to the physical process of molecular diffusion.1 As a chemical reaction occurs, volume and configurational entropy decrease thus slowing the rate of diffusion, which in turn, slows the rate of polymerization after the reaction kinetics have changed from mass-controlled to diffusion-controlled. The slowness of reaction decreases the rate at which the liquid’s physical properties change spontaneously with time. Consequently, a negative feedback is established between the polymerizing reaction and molecular diffusion. This vitrifies a liquid isothermally and brings the polymerization to become unobservable on the experiment’s time scale.1 An increase in the hydrostatic pressure is expected to decrease the rate of polymerization because of a general increase in viscosity on compression of a liquid (or decrease in volume and configurational entropy) as well as an increase in the polymerization extent according to the Le Chatelier principle. But dielectric properties used for studying the rate of polymerization show the opposite; that is, pressure accelerated the polymerization in both linear chains and network structures.2-4 Although counterintuitive, this indicated that the polymerization kinetics remains mass-controlled up to a substantial extent of polymerization, where a decrease in the diffusion coefficient of the reacting species does not affect the polymerization and the increase in their number density per unit volume, which raises the probability of the reaction, overrides the effects of slowing diffusion. In the diffusion-controlled regime of polymerization kinetics, the increase in viscosity dominates the effects of increase in the number density,4-6 and in the intermediate situation of mass and diffusion control, where the consequences of an increase
in the number density compensate for the consequences of an increase in the viscosity, pressure is expected to have no effect on polymerization.4 In addition, the viscosity increase caused by compression of the liquid and the spontaneous viscosity increase on its polymerization are expected to become mutually dependent as the liquid polymerizes toward its vitrification. Here, we report an investigation in which these effects have been carefully examined by using the dielectric measurements technique. In particular, polymerization of four diepoxideamine liquid mixtures has been studied under isothermal conditions. In two cases, the step-polymerization leads to the formation of linear chains and in the other two to a network structure. In each case, the pressure was applied in a single step during the course of polymerization, maintained for a predetermined period, and then removed. The procedure was repeated such that the entire polymerization of a mixture could be studied over multisteps (of subsequent upsteps and downsteps) of pressure between 1 and 200 bar. For comparison, a study of the same four liquids during their polymerization at fixed pressure of 1 and 200 bar is included. The results are then analyzed by a combined formalism of the concepts developed in earlier studies.2-4 In addition to the fundamental interest in diffusion-control reactions, the study is relevant to technological procedures of pressure-molding the thermosetting polymers. Experimental Methods The chemicals aniline (AN), n-hexylamine (HA), ethane-1,2diamine (EDA), and hexane-1,6-diamine (HDA) were of highest purity purchased from Aldrich Chemicals. Diglycidyl ether of bisphenol A (DGEBA) was provided by Shell Chemicals under the trade name EPON 828. Its molecular wt is 380, and its functionality (epoxy group per molecule) is 2. Because of the
10.1021/jp9831076 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/20/1999
3998 J. Phys. Chem. B, Vol. 103, No. 20, 1999 irreversible nature of the phenomena studied here, a new sample was studied for each of the two pressures, 1 and 200 bar, and for the upstep and downstep pressures. Stoichiometric compositions (1 mol of DGEBA with 1 mol of monoamine, and 2 mol of DGEBA with 1 mol of diamine) of four polymerizing liquids were studied: (i) DGEBA-HA; (ii) DGEBA-AN, which form a linear chain polymer structure; (iii) DGEBA-EDA; (iv) DGEBA-HDA, which form a network structure. For linear chain polymers, the n-hexyl group becomes a side group in (i) and the phenyl group does so in (ii). For the network structure polymers, the -(CH2)2- group forms a crosslink in (iii) and a -(CH2)6- group in (iv). An amine chemically reacts completely with a diepoxide in two stages. In the first stage, a linear chain or part of a network structure is formed in the following sequence. One H atom of the NH2 group of the amine combines with the -O- group and forms an -OH group. This opens the terminal epoxide ring, and the N atom becomes covalently bonded to the terminal C atom of the diepoxide molecule. The second H atom of the now more sterically hindered -NH- group in the R-NH(CH2)nCH3 product of the monoamine that produces a linear chain structure (where R denotes the partly reacted diepoxide molecule) reacts similarly with the terminal epoxide group of another diepoxide molecule. (For diamine, which produces a network structure, R-NH-(CH2)n-NH2 reacts similarly.) Thus, by losing its two amine H atoms, one NH2 group links two epoxide molecules by its N atom. The two N atoms of a diamine therefore link four diepoxide molecules, and the group between the N atoms becomes a cross-link. Hence, the number of -OH groups formed is exactly equal to the number of covalent bonds formed between the N and C atoms in the macromolecule. A further reaction known as an etherification reaction is insignificant for a stoichiometric composition but becomes important when the diepoxide is in excess. The liquid mixture may vitrify before the reaction has reached completion on polymerization at low temperatures or may remain a liquid even after complete reaction when polymerization is at high temperatures. The dielectric measurement assembly, the temperature control, and the detail of the hydrostatic pressure assembly and the procedure have been described before.2,3,7 The dielectric permittivity and loss, ′ and ′′ were determined at a fixed frequency of 1 kHz at 30 s intervals during the course of polymerization for all conditions of temperature and pressure. First, measurements were made when the liquid mixture was kept at 1 bar and at 200 bar pressures, and both of these measurements served as a reference. The pressure was step-increased and then stepdecreased at regular intervals during the course of polymerization. In a typical set, the pressure on the liquid mixture was 1 bar. After a duration of 500-1000 s, the pressure was stepincreased to 200 bar. After an equal duration of time, the pressure was step-decreased back to 1 bar, and the procedure was repeated five or six times during the course of polymerization. Thus, the liquid was subjected to a series of up and down pressure steps over a broad range of its viscosity and extent of chemical reaction. Results Figures 1-4 show the plots of the dielectric permittivity, ′, and loss, ′′ ()σmeas/ω0, where σmeas is the measured conductivity, ω the angular frequency, and 0 the permittivity of free space, 8.85 pF/m), of the four polymerizing liquids measured for a fixed frequency of 1 kHz against the polymerization time, t. Each figure contains three plots, one for measurements at a fixed pressure of 1 bar, the second for that at 200 bar, and the third for the upstep-downstep pressures between 1 and 200
Wasylyshyn and Johari
Figure 1. (a) Pressure profile during the polymerization of DGEBAAN at 351.5 K; (b) ′ and (c) ′′ measured for 1 kHz are plotted against the polymerization time for the profile indicated in (a).
Figure 2. (a) Pressure profile during the polymerization of DGEBAHA at 303.7 K; (b) ′ and (c) ′′ measured for 1 kHz are plotted against the polymerization time for the profile indicated in (a).
bar during the course of polymerization. A comparison of the plots in Figures 1-4 shows that the ′ and ′′ for the four liquid mixtures vary with t, qualitatively similarly, with the pressure shifting the curve bodily toward shorter t in all cases, as observed before.2,3 In the upstep-downstep experiments, the magnitude of the expected jog or step discontinuity in the ′ and ′′ data seems to depend on the extent of polymerization. Because of the lack of calorimetric data for similar conditions, this extent is defined here by t. An analysis of the data of the first two sets in terms of the dielectric relaxation time and configurational entropy of the first two experiments would be similar to that in the earlier papers and need not be included.2,4 Discussion An increase in pressure has two effects on polymerization. First, physical which is reversible and restores the liquid to its
Linear Chain and Network Polymerization
J. Phys. Chem. B, Vol. 103, No. 20, 1999 3999 relaxation times; (v) evolution of a second relaxation process; (vi) a decrease in σdc, the dc conductivity. According to the Onsager-Kirkwood-Fro¨hlich equation,14-16
s - ∞ )
Figure 3. (a) Pressure profile during the polymerization of DGEBAEDA at 307.2 K; (b) ′ and (c) ′′ measured for 1 kHz are plotted against the polymerization time for the profile indicated in (a).
( )( ∞ + 2 3
2
)( )
3s 4πNd gµ02 2s + ∞ 3kBT
(1)
where Nd is the number density of the dipoles, kB the Boltzmann constant, µo, the average dipole moment of the species, and g their orientational correlation factor. This equation is used to describe the behavior of polymers containing a variety of dipole moments, distribution of entities of different molecular weights and the unreacted components and plasticizers, and it is used here to approximately describe the complex mixture of reactants and products of polymerization. (It should be noted that “. . . the application of the Fro¨hlich theory to a particular model for a liquid polymer gives a relation very similar to that obtained for associating small molecules in the liquid state”.17) When a change in g and µo on a 200 bar change in pressure is ignored and the change in ∞ due to the optical and vibrational polarization is negligible,
(∂s/∂P) ∝ (∂Nd/∂P)
(2a)
or
s(Pii) ) s(Pi)
[ ] Nd(Pii) Nd(Pi)
(2b)
where Nd(Pi) and Nd(Pii) refer to the number density of the dipoles at pressure Pi and Pii, respectively, and s(Pi) and s(Pii) to the corresponding static permittivity. In terms of the transition state theory applied to dielectric relaxation and viscous flow, the pressure dependence of the relaxation time is expressed as18
(∂ ln τ/∂P)T ) ∆Vτ*/RT
Figure 4. (a) Pressure profile during the polymerization of DGEBAHDA at 303.4 K; (b) ′ and (c) ′′ measured for 1 kHz are plotted against the polymerization time for the profile indicated in (a).
original state after the pressure is removed, as observed in most chemically invariant liquids.7-13 Second, chemical, which is irreversible and observable only when a spontaneous chemical change occurs with time. As they affect also each other in such a manner that a polymerizing liquid’s properties are not recovered on removal of the pressure, it seems appropriate to first discuss them separately. Physical Effects of Pressure through Instantaneous Densification. Analogous to a decrease in temperature, hydrostatic pressure affects the dielectric behavior of a chemically invariant material in at least six ways:1 (i) a change in s, the limiting low-frequency or static permittivity; (ii) a change in ∞, the limiting high-frequency permittivity, which is the sum of the polarizations in the optical and infrared regions and any contributions from a faster relaxation; (iii) an increase in τ, the average relaxation time; (iv) a change in the distribution of
(3)
where ∆Vτ* is the activation volume for dielectric relaxation and R is the gas constant. Since the changes in s, ∞, and the distribution of relaxation times on an increase in P are usually much smaller than the increase in τ, and a second relaxation process has not evolved sufficiently to contribute to the 1 kHz frequency data, an increase in P affects the magnitude of ′ and ′′ mainly by increasing τ. This increase depends on the magnitude of ∆Vτ*, which itself has been found to increase with P, such that the plots of ln τ against P are found to ultimately curve upward.9,12,13 The dc conductivity, σ, of a material also contributes to the measured ′ and ′′, the first through interfacial effects and the second through the σ/ω0 term (0 being the permittivity of free space). As pressure increases the viscosity of a liquid, the mobility of its ionic impurities decreases as does the protonic conduction. The decrease occurs according to an equation similar to eq 3,
(∂ ln σ/∂P)T ) -∆Vσ*/RT
(4)
where ∆Vσ* is the volume of activation for dc conductivity. Chemical Effects of Pressure on Polymerization Kinetics. Polymerization kinetics is usually mass-controlled initially, but as the viscosity increases and the liquid approaches vitrification isothermally, it becomes diffusion-controlled. Thus, the effect of pressure in the two cases differs. As the pressure of a chemically invariant liquid is raised, its shear (and volume)
4000 J. Phys. Chem. B, Vol. 103, No. 20, 1999
Wasylyshyn and Johari
viscosity, η, increases according to the relation18
where ∆Vk* is the volume of activation for the reaction, whose magnitude is usually negative, so that k increases on raising P. ∆Vk* itself is found to depend on the pressure. Combining eqs 8 and 5,
(∂ ln η/∂P)T ) ∆Vη*/RT
(5)
where ∆Vη* is the activation volume for viscosity. ∆Vη* has been found to increase with P nonlinearly as the ln η plots curve upward with increase in P.10 If τ was directly proportional to η, according to the Maxwell’s relation, τ ∝ η, ∆Vτ* will be equal to ∆Vη*. The pressure-induced step increase in η according to eq 5 may be combined with the spontaneous increase in η on isothermal polymerization, and the combined effect may be related to the chemical rate constant, k, representing the polymerization process. The pressure dependence of k may be written as
(∂ ∂Pln k) ) (∂∂ lnln Vk ) (∂ ∂Pln V) + (∂∂ lnln ηk) (∂ ∂Pln η) T
T
T
T
T
(6)
where V is the volume of the liquid. Since βK ) -(∂ ln V/∂P)T is the isothermal compressibility, eq 6 becomes
(∂ ∂Pln k) ) -β (∂∂ lnln Vk ) + (∂∂ lnln ηk) (∂∂lnPη) T
K
T
T
T
(7)
Because a decrease in the volume increases the probability of reaction in the mass-controlled regime, (∂ ln k/∂ ln V)T in eq 6 is negative, causing the first term on the RHS of eq 7 to remain positive, The second term remains negligible because the term (∂ ln k/∂ ln η)T is zero in the mass-controlled regime.5,6 In the diffusion-controlled regime, (∂ ln k/∂ ln η)T is found to be negative, and since (∂ ln η/∂P)T is usually positive, the second term on RHS remains negative in the mass-controlled regime. The magnitude of the (∂ ln k/∂P)T in eq 7 therefore may range from positive to negative values depending upon the mass- and diffusion-controlled regime of the polymerization kinetics. It is worth considering how the magnitude of (∂ ln k/∂P)T changes during the course of polymerization. In the early stages, when the kinetics is mass-controlled and the liquid’s viscosity is low, only the first term on the RHS in eq 7 is significant, (∂ ln k/∂P)T > 0, and k increases with increase in P. Also, as the liquid’s volume decreases spontaneously on polymerization at a constant T and P, βK in eq 7 decreases typically from ∼10-4 bar-1 for the liquid state to ∼10-6 bar-1 for the vitrified state. This means that the positive first term on the RHS of eq 7 decreases as βK decreases with t, thereby increasing the relative significance of the second term. In the latter stages, when η is high and the polymerization kinetics approach the diffusioncontrolled regime, the negative second term on the RHS in eq 7 becomes significant relative to the first term, and its importance increases further as η continues to increase during polymerization. This increase depends on the magnitudes of (i) the term (∂ ln η/∂P)T of eq 5 and (ii) the chemical effect through the term (∂ ln k/∂ ln η)T of eq 7. Taken together, this means that the measured (∂ ln k/∂P)T will be positive in the beginning of polymerization, zero when the increase in k caused by densification alone on raising P is compensated by the decrease in k caused by an increase in η on raising P, and ultimately will become negative when the second term on the RHS of eq 7 exceeds the first term and will remain so until complete polymerization. The pressure dependence of k is considered frequently in terms of the transition state theory. Accordingly, for a masscontrolled kinetics,5
(∂ ln k/∂P)T ) -∆Vk*/RT
(8)
(∂ ln k/∂ ln η)T ) -∆Vk*/∆Vη*
(9)
In the mass-controlled regime, when (∂ ln k/∂ ln η)T or -∆Vk*/ ∆Vη* ≈ 0, it is implied that -∆Vk* , ∆Vη*, and in the diffusion-controlled regime, when (∂ ln k/∂ ln η)T or -∆Vk*/ ∆Vη* > 1, it is implied that ∆Vk* . ∆Vη*. It follows that the plots of -ln k against P would cross the plots of ln η at a certain P. Below this P, the reaction is mass-controlled and above, it is diffusion-controlled. General Effects of Pressure on the Dielectric Behavior. Both the physical and chemical changes alter the pressure’s (upstep and downstep) effect on ′ and ′′, When a change in the distribution of τ with an increase in the extent of polymerization, R, after the main or R-relaxation has sufficiently emerged is justifiably neglected,19-22 ′ may be written in terms of R, up to a certain extent of polymerization, Ri, at ti,
′(Ri) ) [′(R)0)]P1 +
[( )( ) ( )( )] ∫ [( )( ) ( )( )] ∫ [( )( ) ( )( )]
∫0R
1
P2
P1
∂′ ∂s ∂′ ∂τ + ∂s ∂R ∂τ ∂R
P1
∂′ ∂s ∂′ ∂τ + ∂s ∂P ∂τ ∂P
R1
Ri
R1
dR +
dP +
∂′ ∂s ∂′ ∂τ + ∂s ∂R ∂τ ∂R
dR (10) P2
and, in terms of the polymerization time, t, up to ti
′(ti) ) [′(t)0)]P1 +
∫0t
[(( )( ) ( )( ))( )] ∫ [( )( ) ( )( )] ∫ [(( )( ) ( )( ))( )] ∂′ ∂s ∂′ ∂τ ∂R + ∂s ∂R ∂τ ∂R ∂t
1
P2
P1
∂′ ∂s ∂′ ∂τ + ∂s ∂P ∂τ ∂P ti
t1
dt +
P1
dP +
t1
∂′ ∂s ∂′ ∂τ ∂R + ∂s ∂R ∂τ ∂R ∂t
dt (11)
P2
where P is the pressure. Correspondingly, ′′ can be expressed as
′′(Ri) ) [′′(R)0)]P1 +
[( )( ) ( ∫ [( )( ) ( ∫ [( )( )
∫0R
1
P2
P1
( )]
∂′′ ∂s ∂′′ ∂τ 1 ∂σ + + ∂s ∂R ∂τ ∂R ωo ∂R
P1
∂′′ ∂s ∂′′ ∂τ 1 ∂σ + + ∂s ∂P ∂τ ∂P ωo ∂P
R1
Ri
R1
)( ) )( )
( )]
or, in terms of t,
dP +
( )]
1 ∂σ ∂′′ ∂s ∂′′ ∂τ + + ∂s ∂R ∂τ ∂R ωo ∂R
( )( )
dR +
dR (12) P2
Linear Chain and Network Polymerization
′′(ti) ) [′′(t)0)]P1 +
[(( )( ) ( )( ) ∫ [( )( ) ( )( ) ∫ [(( )( ) ( )( )
∫0t
J. Phys. Chem. B, Vol. 103, No. 20, 1999 4001
( ))(∂R∂t )]
∂′′ ∂s ∂′′ ∂τ 1 ∂σ + + ∂s ∂R ∂τ ∂R ωo ∂R
1
P2
P1
ti
t1
( )]
∂′′ ∂s ∂′′ ∂τ 1 ∂σ + + ∂s ∂P ∂τ ∂P ωo ∂P
dt +
P1
dP +
t1
( ))( )]
1 ∂σ ∂R ∂′′ ∂s ∂′′ ∂τ + + ∂s ∂R ∂τ ∂R ωo ∂R ∂t
dt (13)
P2
In eqs 10-13, σ is the dc conductivity and the other quantities are as described before. The limits of integration here are P1 (1 bar or Pi) to P2 (200 bar or Pii) and from t1 to ti and R1 to Ri. The second integral in eqs 10-13 represents the net physical effect of the pressure step, and the first and third integrals represent the net chemical or polymerization effect. The physical effect, which is instantaneous under most conditions after the pressure step, alters the static and dynamic properties of the liquid, and the chemical effect, which is time-dependent, alters the polymerization kinetics depending qualitatively upon whether the kinetics are mass-controlled or diffusion-controlled. For the first and second sets of experiments at P ) 1 and 200 bar, the second and third integral terms make no contribution to the ′ and ′′ data. In the third set of experiments with upstep-downstep pressure, all integral terms contribute to ′ and ′′, but, as the (∂′/∂s)(∂s/∂P) term is negligible in comparison with the (∂′/∂τ)(∂τ/∂P),9,23 all terms involving ∂s may be neglected in comparison with those involving ∂τ. This leads to
′(t) ) [′(t)0)]P1 +
∂τ ∂R ∫0t [(∂′ ∂τ )(∂R)( ∂t )]P
dt +
1
1
t ∂′ ∂τ ∂R ∂τ dP + ∫t [( )( )( )] ∫PP [(∂′ ∂τ )(∂P)]t ∂τ ∂R ∂t P 2
i
1
1
1
dt (14) 2
At times before gelation during polymerization, σ is large and important, but after gelation, it is negligible,1 as is its change with P.2 Thus, after gelation in the third set of experiments,
′′(t) ) [′′(t)0)]P1 +
∂τ ∂R ∫0t [(∂′′ ∂τ )(∂R)( ∂t )]P
dt +
1
1
i
1
1
dependence of βK, and according to eq 2b, when R ) Ri, s,ii ) s,i(Nd(Pii)/Nd(Pi)). Changes in τ and σ with the pressure’s upstep and downstep occur according to eqs 3 and 4. Polymerization changes the magnitude of s, ∞, τ, and σ with time so that in the upstep-downstep experiments the state of the sample continuously changes differently in each subsequent step. According to the Le Chatelier principle, pressure favors the formation of a denser (polymeric) product, which means that the ultimate extent of polymerization at chemical equilibrium is greater at high pressures than at low. But the rate at which polymerization proceeds toward this ultimate extent depends on whether the polymerization is in the mass- or diffusion-controlled regime.5,6 In the absence of an appropriate relation between R and t, the following empirical equation has been used to describe the polymerization kinetics.26
1
t ∂′′ ∂τ ∂R ∂τ dP + ∫t [( )( )( )] ∫PP [(∂′′ )( ) ] ∂τ ∂P t ∂τ ∂R ∂t P 2
Figure 5. (a) Pressure profile used for simulation. (b) Simulated R is plotted against the polymerization time, and (c) τ is plotted against t. For both R and τ, the dotted lines correspond to polymerization at 1 bar and the dashed lines that at 200 bar. The continuous lines represent the result of the pressure profile as indicated in (a). Parameters for the simulation are given in the text.
dt (15)
R ) 1 - exp(-k′tm)
(16)
2
We use eqs 10-15 for a further discussion of the physical and chemical effects of pressure on ′ and ′′ of a polymerizing liquid. Simulation of E′ and E′′ on Pressure’s Upstep and Downstep. The time-dependent changes in ′ and ′′ following a pressure’s step involve two effects. (i) The equilibrium dielectric properties, s, ∞, τ, and σ, may become time-dependent on the experiment’s time scale when the viscosity exceeds 1011 Poise (or τ > 10 s). (ii) The rate of polymerization changes. According to eq 1, a pressure step alters the magnitude of (s - ∞) as Nd, g, and µo change. In relative terms, the effect of P on ∞, which is due to the increase in the optical refractive index and infrared polarizability, has been found to be less than 0.1%/ kbar2,12,23,24 and therefore is negligible in comparison with changes in Nd, g, and µo. As gµo2 varies by up to ∼10% for 4 kbar23 and the maximum pressure used here was 206 bar, (dgµo2/ dP) is also negligible compared to (dNd/dP). Hence, we deduce that, (ds/dP)R is proportional to (dNd/dP)R or to -(dV/dP)R. The change in (dV/dP)R with P is expressed in terms of the P
where m and k′ are empirical parameters whose magnitudes determine R at a given t. Since an increase in P increases k′, which determines the rate of polymerization generally in the mass-controlled regime, it is expected that k′ will similarly increase with P. (Note that k′ is not identical to k of eq 8 but is directly related to it, and when m ) 1, k ) k′.) The increase in R on increase from k′i (at Pi) to k′ii (at Pii) may be calculated from eq 16 by assuming a pressure-independent m ) 1 and therefore substituting k′ by k. For a qualitative simulation of the shape of the observed ′ and ′′ curves in Figures 1-4, we use, ki ) 10 µs-1 and kii ) 14 µs-1, and a P step at 5 ks intervals, as shown in Figure 5a. The plots of R calculated for a fixed ki and kii using eqs 8 and 16 are shown by the dotted and dashed curves, respectively, in Figure 5b. The two curves correspond to a fixed Pi ()1 bar) or Pii ()200 bar) from the beginning of the experiment. The onset of diffusion control is expected to decrease the slope of the plots at longer time more than is shown in Figure 5b. Every upstep from Pi to Pii in the early stages of polymerization increases the polymerization rate as ki increases
4002 J. Phys. Chem. B, Vol. 103, No. 20, 1999
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to kii and remains so for the duration of Pii. Every downstep to Pi , after a certain time interval, decreases the polymerization rate as kii decreases to ki and remains so for the duration of the step. By taking these into account, the curve corresponding to the upstep-downstep in Figure 5a may be calculated from the equation
R(t) )
t ∂R dt + ∫t ( )Pii dt ∫0t (∂R ) ∂t P ∂t 1
N
i
(17)
1
and is shown in Figure 5b. To elaborate, at Pi, R increases at a rate corresponding to ki up to t1 (and up to R1). At t1, Pi is upstepped to Pii and the polymerization proceeds up to t2 (from R1 to R2) with a rate constant kii at Pii. Thus, (∂R/∂t)T before t1 is less than (∂R/∂t)T after t1. When Pii is downstepped to Pi, R2 increases according to the rate constant. At t2, R increases from R2 at a rate ki, and eq 16, but the rate of its increase differs from that at the immediately preceding Pi. The curve for R in the upstep-downstep experiment continuously shifts toward the dashed-line curve for a fixed Pii in Figure 5b, but cannot reach it because part of the polymerization occurs with a lower k. (The increased extent of polymerization at an internal equilibrium at Pii required by the Le Chatelier principle is likely to be too small to compensate for it.) This R-t profile may now be combined with the dielectric parameters to simulate the net changes in ′ and ′′. For determining the change in Nd, we consider the decrease in volume on polymerization, which being as much as 10%,25 is related to R by
V(R) ) Vo + (Vf - Vo)R
(18)
where Vo is the volume of the unpolymerized liquid (R ) 0), and Vf is that of the polymer (R ) 1). P and V are related by isothermal compressibility, βK, which decreases on polymerization. To a first approximation, βK(R) is proportional to R such that
βK(R) ) βK,f + (βK,f - βK,o)R
(19)
where βK,o and βK,f are the values of βK at R ) 0 and R ) 1, respectively. Combining eqs 18 and 19,
Vii(R) ) Vi(R)[1 - βK(R)∆P]
(20)
where Vi(R) and Vii(R) are the volumes at the fixed extent of reaction R at Pi and Pii, respectively, and ∆P ) Pii - Pi. The consequence of eq 20 is that for each P step, the change in V depends on R through the R dependence of βK(R). During the interval between the upstep and downstep, the volume would decrease more rapidly on polymerization than before the upstep, because kii > ki. For an upstep from Pi to Pii at t1, Vi(R) would step-decrease by an amount Vi(R)βK(R)∆P to Vii(R). From t1, Vii(R) would decrease Vii(Rι) at a rate corresponding to kii. In the downstep, Vii(R1) would increase to Viii(R1) by an amount Viii(R)βK(R)∆P and then decrease according to ki, and eq 16, toward Viii(R2). For our calculations, we use the reasonable values of βK,o ) 10-5 bar-1, βK,f ) 10-6 bar-1, and ∆P ) 200 bar to calculate the volume for each upstep and downstep to determine the change in s through the change in Nd, according to eq 2. The limiting low-frequency permittivity, s, of a molecular liquid decreases as the number density of dipoles, Nd, the orientational correlation factor, g, and the vapor-phase dipole moment, µo, according to eq 1 vary as R increases during
polymerization. For describing the decrease in s with increase in R, we write
s ) s,o + (s,f - s,o)R
(21)
where s,o refers to s at R ) 0 and s,f to that at R ) 1. The rate of change of s with T and P depends on the rate of change of R with t and P through k of eq 6. As the R-relaxation process evolves on polymerization, its ∞ decreases with an increase in R and t.24 This decrease is relatively small at those R values where the relaxation’s effect on ′ and ′′ for a fixed frequency of 1 kHz becomes significant and therefore may be ignored. In summary, s and ∞ decrease with t, through R which increases according to the magnitude of k, and the latter itself depends on P. On upstep from Pi to Pii at a time, t1(R1), the step increase in from s,i to s,ii is mainly due to the increase from Nd,i to Nd,ii. Their effects on ′ and ′′ are considered negligible here but are included here for completeness of the treatment and for use in the future for cases where the changes in s and ∞ with R are relatively large. The most sensitive property to the process of polymerization is molecular diffusivity, or equivalently τ. It decreases, or τ increases, by nearly 14 orders of magnitude with t, as R f 1, according to an empirical equation,1,20,26-28
τ ) τo exp(SR)p
(22)
where τo is the average relaxation time at R ) 0 or t ) 0, p is the reaction parameter, and S ) ln(τ∞/τo) with τ∞ the average relaxation time at R ) 1. Both τo and τ∞ increase with the step increase in P according to eq 3, but as τ∞ increases more than τo, S increases on increase in P. The change in p with P is not known and cannot be appropriately deduced. For simplicity of treatment, p is taken as unity and independent of P, which makes the plot of ln τ against R linear with a slope of S ) ln(τ∞/τo) and endpoints of τo and τ∞. The rate of change of ln τ with t is related to the rate of change of R with t, according to eqs 22 and 16. In the latter equation, τ increases with t according to the magnitude of k (m ) 1), τo, and τ∞. As an upstep increases k, τo, and τ∞, τ can be calculated for ki at Pi and kii at Pii, τo,i, τo,ii, τ∞,ii, τ∞,ii, Si, and Sii and a constant p, using eqs 3, 8, 16 and 22. By using τo,i ) 1 ns and τ∞,i ) 1 Ms in eq 22, the same ki as for the calculation of R from eq 16, and ∆V*τo ) 12 mL/mol and ∆V*τ∞ ) 42 mL/mol in eq 3, τi and τii were calculated for Pi ) 1 bar and Pii ) 200 bar and are plotted against t as dotted and dashed lines, respectively, in Figure 5c. The full line curve in Figure 5c was calculated for the upstep-downstep profile in Figure 5a using the equation
τ(t) ) τo,i +
∂τ ∂R ∫0t [(∂R )( ∂t )]P dt + ∫PP [(∂P∂τ )]t 1
dP +
ii
i
i
1
∂τ ∂R ∫t t [(∂R )( ∂t )]P N
1
dt (23) ii
To elaborate, during polymerization from the beginning to t1 at Pi, ln τ increases according to eq 22 with τo ) τo,i and τ∞ ) τ∞,i and eq 16 with k ) ki. On upstep at t1 (and R1), from Pi to Pii, the step increase in τ at R1 is due to the increase of τo,i to τo,ii and τ∞,i to τ∞,ii, respectively. From t1 (R1), ln τ changes at an increased rate according to τo,ii, τ∞,ii in eq 22 with increase in ki to kii. On downstep at t2 (and R2) from Pii to Pi, the step decrease in τ is due to the decrease of τo and τ∞ to values of τo,i, and τ∞,i, respectively. From t2 (and R2) to time tN, τ increases according to eq 22 with τo ) τo,i, τ∞ ) τ∞,i, and ki in eq 16.
Linear Chain and Network Polymerization
J. Phys. Chem. B, Vol. 103, No. 20, 1999 4003
Thus, the step change in τ with P, (∆ log τ/∆P)tN, is positive for ∆P > 0 and negative for ∆P < 0, and like the R curve, the ln τ curve in Figure 5c continuously shifts toward the dashed line curve for Pii but does not reach it because τ at an internal equilibrium at Pii is greater than τ at Pi, as R is greater at Pii than at Pi. Finally, we consider the change in dc conductivity σ, on upstep and downstep. It decreases on polymerization from an initial value, σo at t ) 0 (R ) 0) as the diffusivity of the impurity ions decreases, the population of zwitterions is reduced28 and the proton transfer along the intermonomer H-bond becomes less frequent as covalent bonds form. The change in σ may be discussed similarly to changes in s and ∞, i.e., in terms of the upstep-downstep’s effect on k of eq 16. An upstep increases k, and σ decreases more rapidly at Pii than at Pi. The decrease in σ with t during the network formation has been expressed as1,20,29
σ ) σo
[ ] tgel - t tgel
x
(24)
where σ0 is the dc conductivity at t ) 0 or R ) 0, tgel is the time at which the cross-linked gel forms, and x is the critical exponent. Equation 24 has been used to describe the σ of a variety of cross-linking epoxide-diamine mixtures, whose tgel and Rgel (the value of R at tgel) have been obtained from the dielectric and shear wave propagation measurements,30 and it is shown that σ of a diepoxide-monoamine polymerization in which no cross-links form disobeys eq 24.20 Since it is derived from statistical considerations,31-33 Rgel should be independent of the rate of polymerization, and thus P is not expected to affect Rgel, although it decreases tgel. For a fixed Rgel, τgel,i and τgel,ii may be determined from eq 16 for values of ki at Pi and kii at Pii for our upstep-downstep experiments. From eq 6, and τgel,i and τgel,ii, the decrease in σ with t may be determined for both Pi and Pii. For qualitative simulation, the parameters chosen were Rgel ) 0.5, s,o ) 10, s,f ) 8, σi ) 50 nS/m, σii ) 45 nS/m, and ∞ ) 4. For convenience of qualitatively simulating the effects of upstep and downstep on ′ and ′′, we ignore the distribution of relaxation times and use the single relaxation time (Debye model) equation.
∆ 1 + ω2τ2
(25)
∆ωτ σ + 2 2 ω 1+ω τ o
(26)
′ ) ∞ + and
′′ ) ∞ +
where ∆ ) s - ∞ and ω is the angular frequency. The ′ and ′′ were calculated as a function of polymerization time for a fixed ω ()1 krad/s) from changes in the equilibrium properties s, ∞, τ, and σ through change in k from eq 6. The dotted line was calculated for ′ and ′′ with ki ) 10 µs-1 in eq 16 for 1 bar and the dashed line with kii ) 14 µs-1 in eq 16 for 200 bar. Because kii > ki, the dielectric features of the plots of ′ and ′′ evolve more rapidly for the calculated curves corresponding to 200 bar (Pii) than for that at 1 bar (Pi). For the upstep-downstep profile in Figure 6a, the calculated ′ and ′′ values are shown by dots in parts b and c of Figure 6. The discontinuities in the values at each of the pressure steps at time, tN, arise from the instantaneous physical changes in s, τ, and σ, through the relationships described by eqs 1-8. After
Figure 6. (a) Pressure profile used for the simulation; (b) the simulated ′ is plotted against the polymerization time, t, and (c) the simulated ′′ is plotted against t. For both ′ and ′′, the dotted lines correspond to polymerization at 1 bar, and the dashed lines that at 200 bar. The filled circles are the simulated values for the pressure profile, as indicated in (a). Parameters used for simulation are given in the text.
the discontinuity, the slower change in ′ and ′′ in parts b and c of Figure 6 are described by eqs 16-24, with t-dependent (therefore, R-dependent) values of τ, σ, and s at the relevant pressures. High pressure causes ′ and ′′ to evolve more quickly than for lower pressures. The simulated effects of upstep-downstep pressure on the evolution of the dielectric properties in Figure 6 qualitatively agree with those observed in Figures 1-4. In the plots of ′ against t in Figure 6b, the step response of ′ to pressure is a result of a finite magnitude of (∆s/∆P)t, through (∆Nd/∆P)t from eq 2, with Vi(R)/Vii(R) ) Nd,ii/Nd,i and (∆τ/∆P)t from eq 3. Similarly, in Figure 6c, the step response of the simulated ′′ to pressure during polymerization is a result of (∆s/∆P)t, through (∆Nd/∆P)t from eq 2, (∆τ/∆P)t from eq 3, and (∆σ/ ∆P)t from eq 4. Thus, the overall changes in both ′ and ′′ observed experimentally can be formulated in terms of the concepts developed here. Although simplifying assumptions were made in order to more easily calculate and simulate the observed behavior. Simultaneous Changes in Temperature and Pressure. We now consider the effect of temperature change concomitant with the pressure upstep and downstep during the polymerization. In a closed system, a step increase in P is accompanied by a step increase in T, and thereafter T decreases asymptotically to the equilibrium value at that P. Similarly, a step decrease in P is accompanied by a step decrease in T, and thereafter T increases to the equilibrium value. These changes in T were observed here and the typical effects of upstep and downstep of P on T are shown in the top two parts of Figures 7 and 8. Although the combined effects of P and T can be incorporated in the simulation, the combination becomes too complicated to be useful here, and since the observation of a P-T profile is incidental to our concern, its qualitative discussion may be sufficient. The consequences of the P-T profile are also of course evident in Figures 1-4, and the enlarged plots of ′′ in the early stages (right-hand side) and the later stages (left-hand side) of polymerization corresponding to the pressure’s upstep and
4004 J. Phys. Chem. B, Vol. 103, No. 20, 1999
Wasylyshyn and Johari decrease in T on pressure’s downstep produces an exactly opposite effect to the vertical increase in T on pressure’s upstep. In the early stages of polymerization, σ is significantly high and an increase in it has an effect on ′′ opposite to that of increase in τ. This makes the net effect relatively less. The slightly nonmonotonic change in ′′ on pressure’s upstep and downstep seen in detail on the right-hand panels of Figures 7 and 8 is due to the compensatory effect of the two. In the latter stages of polymerization, σ is negligibly small and any change in it is inconsequential. Thus, only the effects of P-T profiles on τ are important. Since the compensatory effect of σ on ′′ is now absent, the nonmonotonic change in ′′ becomes pronounced in the later stages of polymerization, as seen in detail on the right-hand side panels of Figures 7 and 8, as well as in Figures 1-4. It may be noted that part of the reason for the pronounced effect of the P-T profile on the latter stage of polymerization is likely to be also due to the greater sensitivity of τ to T , when the state of the liquid is already highly viscous or when τ is already long. Conclusions
Figure 7. Details of a pressure profile, the consequent temperature changes, and the changes in ′′ with time. The left-hand panel is for the early stages of polymerization, and the right-hand panel during the later stages of polymerization of a DGEBA-HA mixture at 303.7 K.
The general response of ′ and ′′ to pressure may be described in terms of an equilibrium effect and a time-dependent effect, namely, (i) an increase in Nd and τ and a decrease in σ on a step increase or step decrease of pressure, causing ′ and ′′ to change accordingly, and (ii) a time-dependent change in the polymerization rate on compression and decompression. The change in temperature concomitant with a change in the pressure of a closed system has significant effect on the time-dependent dielectric behavior during the polymerization of liquid. This change is consistent with anticipated and already verified effects of temperature. References and Notes
Figure 8. Details of a pressure profile, the consequent temperature changes, and the changes in ′′ with time. The left-hand panel is for the early stages of polymerization, and the right-hand panel during the later stages of polymerization of a DGEBA-EDA mixture at 307.2 K.
downstep are shown in the bottom part of Figures 7 and 8. The effect is more pronounced on both ′ and ′′ for the linear chain polymerization shown in Figures 1, 2, and 7 than on those for the network polymerization shown in Figures 3, 4, and 8. Since the effects are qualitatively similar, a general discussion seems appropriate. The vertical rise in T on pressure’s upstep increases σ, the dc conductivity, and lowers τ, the two most significant properties. The former raises ′′ and the latter lowers it, for a fixed frequency. The increase in P that affected it decreases σ and increases τ, which partly cancel the effects of the vertical rise in T. As T decreases asymptotically toward the equilibrium value, and P remains constant, the remaining effect of increased T gradually vanishes, and the effect of P persists. The vertical
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