Rate equation, cure

Differential Scanning Calorimetry (DSC) This is by far the widest utilized technique to obtain the degree and reaction rate of cure as well as the specific heat of thermosetting resins. It is based on the measurement of the differential voltage (converted into heat flow) necessary to obtain the thermal equilibrium between a sample (resin) and an inert reference, both placed into a calorimeter [143,144], As a result, a thermogram, as shown in Figure 2.7, is obtained [145]. In this curve, the area under the whole curve represents the total heat of reaction, AHR, and the shadowed area represents the enthalpy at a specific time. From Equations 2.5 and 2.6, the degree and rate of cure can be calculated. The DSC can operate under isothermal or non-isothermal conditions [146]. In the former mode, two different methods can be used [1] ... 

Substituting Equation 8.2 and Equation 8.3 into Equation 8.1, the general expression for the reaction rate of curing polymers is obtained,... 

To obtain the cure kinetic parameters K, m, and n, cure rate and cure state must be measured simultaneously. This is most commonly accomplished by thermal analysis techniques such as DSC. In isothermal DSC testing several different isothermal cures are analyzed to develop the temperature dependence of the kinetic parameters. With the temperature dependence of the kinetic parameters known, the degree of cure can be predicted for any temperature history by integration of Equation 8.5. 

Deng and Martin (1996) also showed the necessity of including a diffusional resistance in the rate equation for the cyclotrimerization of dicyanates, well before vitrification. They observed a significant decrease in the diffusion coefficient from conversions of about 0.40, using dynamic dielectric analysis. They could fit experimental kinetic data in the whole conversion range using Eq. (5.50). Experimental values of the decrease in the diffusion coefficient with conversion were used to estimate kd for different cure temperatures. 

Also important is that the induction period, which was about 3 min in the ESR experiment, was found to increase to about 30 min in isothermal DSC scans performed at the same cure temperature (Tollens and Lee, 1993). This is possibly due to the presence of dissolved oxygen (coming from air) in the DSC samples. Oxygen is a known inhibitor of the UP-S free-radical polymerization. This is a very important fact rate equations determined... 

Figure 5.18 shows the overall conversion of NCO groups as a function of time for uncatalyzed samples cured at three different temperatures. Points are experimental values, while full curves are predicted results using the kinetic parameters derived in the adiabatic analysis. The predictive capability of the rate equation is very good, in spite of the strong hypothesis regarding the absence of substitution effects. 

The cure kinetics were determined using differential scanning calorimetry (DSC). The following second-order rate equation could adjust the experimental results obtained for formulations containing 5-15 parts of hexa ... 

The first attempt to relate changing dielectric properties to kinetic rate equations was by Kagan et al.S9), working with a series of anyhydride-cured epoxies. Building on Warfield s assumed correlation between d log (g)/dt and da/dt, where a is the extent of epoxide conversion, they assumed a proportionality between a and log (q), and modeled the reaction kinetics using the equation... 

Solution of the Rate Equation for Cure. Hie rate at which... 

The kinetics of the cure of ATS at 130°C in nitrogen determined by various methods are shown in Figure 1. The dashed curve represents the prediction based on kinetic parameters determined by analysis of scanning DSC data by the method of multiple heating rates described previously (8). This method utilizes a rate equation of the form ... 

Kinetic Model for Curing Reaction and Comparison Between Predicted and Measured Degree of Cure. We modeled the epoxy curing reaction for all H/R studied, using the n-th order rate equation (1.1) substituting for the rate constant k in (1.1) the expression from the Arrhenius equation in (3.4.1)... 

Although the sulfur vulcanization of the rubber was a complex chemical process "Equation 3", we found that the overall rate of cure was given by a single first-order reaction ... 

The equation is a simple case of a mechanistic model. Models such as this may give better predictions but may not always apply because of the complexity of the reactions. Phenomenological models arc expressed by simple rate equations which ignore the details of the reaction. Phenomenological models are typically used to follow cure rates in polymeric systems which are difficult to follow by chemical analysis. This is because reaction products become insoluble during the course of the reaction and, consequently, are not detected in an analysis of the solution. 

The rate coefficients for the secondary-amine reactions were found to be only 17% of those for the primary-amine reaction, thus explaining the residual secondary amine found at the end of cure. This equation was found to explain the development of the main crosslinking site, namely the tertiary-amine site formed on the DDS, corresponding to network interconnection. However, overlaid with the rate equation for chemical conversion that implies that all reagents are accessible to one another is the effect of the development of the network so that the reactions become diffusion-controlled. This is of interest since this means that the rate coefficients now reflect the chemorheology of the system, not just the chemistry. Thus, if and represent the rate coefficients for diffusion and chemical control, the measured rate coefficient, k, will be given by (Cole et ai, 1991)... 

Equation 3.3, representing the rate of cure, is thus written in terms of temperature ... 

The empirical rate equation proposed by Kamal [38] is applied for the chemically controlled reaction rate of any epoxy resin cure showing auto-catalytic behaviour ... 

An analogous approach has been applied to the epoxy-amine system. The three sets of parameters were derived one set for the chemical rate equation [Eq. (14)], one set for the diffusion rate constant according to Eq. (26), and one set for the T — x relation [Eq. (27)]. As seen in Figure 2.22, the experimental and the calculated DF profiles agree very well for the quasi-isothermal cure at reaction temperatures ranging from 25 to 100°C. 

This is the basic equation of cure for materials of this type, although a few systems do show modest deviations. Values of material properties that control the rate of cure are shown in Table 1. 

The boundary conditions that may be used with the thermochemical module include specified boundary temperature, convective heat transfer or no heat transfer (adiabatic). Different conditions (e.g., different HTCs) can be applied to each element as desired. Either explicit or implicit techniques may be chosen to solve the heat transfer (Eq. [13.1]) and cure rate equations. Using either technique, these two equations are uncoupled during each solution time-step. This approach facilitates a simplified and modular solution procedure and is sufficiently accurate if small time steps are used. 

Yang et al. was used Seo model to correlate of imidization with time [28]. In Seo s approach the rate constant was proposed as k(t) = b x sech(-at). Inserting into the first-order rate equation, the relationship between the degree of imidization and curing time is obtained as ... 

Isothermal Method 1. This method capitalizes on the ability of DSC to simultaneously monitor both the conversion and the rate of conversion over the entire course of the cure reaction. This allows direct use of derivative forms of the rate equation, such as Eq. (2.86), which are necessary for kinetic analysis of autocatalytic reactions such as epoxy-amine. Experimentally this method is well suited to autocatalytic reactions that do not reach maximum rate until later in the reaction after the instrument has achieved thermal equilibrium. Even so, at high temperatures a significant portion of the reaction can take place before the calorimeter equilibrates and go unrecorded. Widmann (1975) and Barton (1983) have proposed a means to correct for such unrecorded heat by rerunning the experiment on the reacted sample, under the same conditions, to obtain an estimate of the true baseline and the unrecorded heat that should be added to the measured heat, as illustrated in Fig. 2.68. Note that this system appears to follow nth-order kinetics where the maximum reaction rate occurs at f = 0. For the sample shown, Widmann reports that 5% of goes... 

Isothermal Method 2. This method is necessary to obtain cure data at low temperatures where the rate of heat evolution is too small for method 1 to be reliable. It is also recommended for nth-order reactions where the maximum rate of cure occurs at f = 0, and to obtain simultaneous Tg and conversion data to construct Jg-conversion plots. The conversion-time data can be fit to integrated forms of the rate equation, such as Eqs. (2.83)-(2.85). Several samples are cured isothermally, for example, in an oven, in the calorimeter or at ambient temperature, for various times until no additional curing can be detected. The samples are subsequently scanned in the DSC at a fixed heating rate, from which Tg and the residual heat of cure (A//res) the heat evolved during completion of the reaction, are measured, as illustrated in Fig. 2.69. 

Lapique and Redford (2002) used differential scanning calorimetry to measure rates of cure for the epoxide paste-adhesive Araldite 2014, finding E to be 34.6 kj mol . The equation also applies to other time-dependent processes such as diffusion and viscous flow. 

Generally the oxidant is compounded in one part of the adhesive, and the reductant in the other. Redox initiation and cure occur when the two sides of the adhesive are mixed. There also exist the one-part aerobic adhesives, which use atmospheric oxygen as the oxidant. The chemistry of the specific redox systems commonly used in adhesives will be discussed later. The rates of initiation and propagation are given by the following equations ([9] p. 221). 

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