Model fit methods

The fundamental theory of thermokinetic cure modelling was presented in detail within submission seven. The aim of thermokinetic cure modelling for the present application was to fit a mathematical model to the adhesive reaction rate experimental data that can then be used to predict curing rate for a given thermal profile. The advantage of this process is that should the thermal cure profile change, it is only necessary to re-apply the model, rather than repeat all of the experimental data. Further, using the model, a thermal profile could be designed to provide a specified joint degree of cure during manufacture based on the thermal history of the assembly within the manufacturing process. If the relationship between degree of cure and mechanical properties can be established the required degree of cure for a specified mechanical property can be identified. Thus, the cure model can be used to minimise cure thermal cycle times to achieve the minimum required handling strength and mechanical properties required during manufacture, contributing to manufacturing efficiency gains.

113

Netzsch Thermokinetcs software was used to perform the cure modelling, benefiting from a library of different model types and the ability to fit and optimise multiple steps for a given reaction of differing configurations. The process fundamentals build on the assumption that the reaction rate (𝑑𝑎

𝑑𝑡) is described by two separate variables; 𝑑𝑎

𝑑𝑡 = 𝑘(𝑇). 𝑓(𝑎) where 𝑘(𝑇) is

a temperature dependant rate constant given by Arrhenius relationship and 𝑓(𝑎) a reaction model which expresses the rate of a reaction as a function of the extent of the reaction (77).

Arrhenius’s relationship is expressed by; 𝑘(𝑇) = 𝐴𝑒−𝐸𝐴𝑅𝑇 , where 𝑘 = reaction rate, 𝐴 a pre-

exponential factor, R the universal gas constant, T temperature and 𝐸𝐴 activation energy. DSC

data was collected for each adhesive model produced using a dynamic scanning method of at least four heating rates, for example 2.5, 5, 10 & 20 °c/minute. Data was normalized by sample mass and a linear baseline subtracted.

Initially model free analysis using the Friedman method (102) was applied to the DSC data to identify the fundamental Arrhenius constants 𝐸𝐴 and A, providing starting parameters for the

model fitting. In the majority of cases multi step models were identified as the most appropriate fit for the adhesives investigated. The Netzch software uses a 6th _{degree Range} Kutta method to optimise the model parameters for each step and model selection (102). The deviation of model response to the collected DSC data was compared and the most accurate fit selected. A summarised example of the model fitting to Lohmann 10400 SBF film is presented, with the process being similar when applied to other products. A more detailed process description of the process is presented within portfolio submission seven.

A model free Friedman analysis was initially applied to 10400 SBF film, identifying start parameters for 𝐸𝐴 and A. Subsequently a nonlinear multiple regression was used to fit various

models and step combinations to the experimental data. The optimum model fit was found to be a two-step process, with first step modelled by a Prout-Tompkins autocatalytic reaction of the form; 𝑑𝑎_𝑑𝑡 = 𝐴𝑒−𝐸𝐴𝑅𝑇 × 𝐸𝑛 × 𝑃𝑎 where E = concentration of educt and P = concentration

of product the optimised model parameters are presented in Table 31. The second step which followed the first was modelled by an nth _{order step of the form; 𝐴𝑒}−𝐸𝐴_{𝑅𝑇 × 𝐸}𝑛_.

114

Step Parameter Optimum value

1 log 𝐴 9.47 1 𝐸𝐴 87.29 1 Reaction order, n 1.07 1 Exponent, a 0.62 2 log 𝐴 2.95 2 𝐸𝐴 39.94 2 Reaction order, n 1.52 1 Foll Reaction.1 0.34

Table 31 Optimised parameters two-step model fit 10400 SBF

The fitted two-step model along with collected DSC data is shown in Figure 79 it can be seen that there is a high degree of correlation between the experimental data and the fitted model. To validate the model, the degree of conversion vs isothermal temperature data was compared to Isothermal DSC measurements of the same adhesive shown in Figure 80, collected using the method discussed in section 4.4.1.

Figure 79 Netzch Thermokinetics model fit 10400 SBF two-step model. Step 1 Prout- Tomkins autocatalytic, step two nth _{order. Solid lines represent model, symbols}

experimental data.

It can be seen in Figure 80 that a very similar profile is produced between the model fit data and the isothermal calculated data. There is however a significant time lag between the model data and collected data. This was attributed to the ramp rate of the DSC deviating from the manufacturers stated value of 300 °c/minute and thermal conduction within the crucible and adhesive, although this would require further work to validate this conclusion. Based on the high degree of curve correlation the model was accepted as accurately representing the curing process. Finally, the model was used to predict the evolution of cure over the induction

115

cycle used within section 4.3.5, Figure 70. Following a 500 °c/minute ramp to 175 °c followed by an isothermal dwell. This enabled the relationship to be identified between the SLS strength upon CFRP/AL substrates at 50 °c test temperature with degree of conversion as shown in Figure 81. 0 100 200 300 400 500 600 0 10 20 30 40 50 60 70 80 90 100 Deg re e o f c o n v e rs io n / % time /s Collected data 125 °C Collected data 150 °C Collected data 175 °C Collected data 200 °C Model prediction 125 °C Model prediction 150 °C Model prediction 175 °C Model prediction 200 °C

Figure 80 Comparison between collected isothermal DSC data and model fit data for 10400 SBF adhesive 20 30 40 50 60 70 80 0 2 4 6 8 10 12 14 SLS s trength at 50 o c (CF RP/A L) / MPa Degree of conversion / % SLS strength model fit

95 % Prediction interval

Figure 81 Relationship between 10400 SBF SLS strength and degree of conversion

Relationships such as Figure 81 could be used to enable optimal thermal cycle times to be designed, in order to generate a specified component handling SLS strength or other mechanical properties such as peel strength or creep behaviour. Extension could also be made to encompass consideration relating to substrate combinations and test temperatures. Thus, manufacturing cycle time can be reduced as well as the required energy input during a pre-

116

cure stage. Further evaluation of the manufacturing thermal cycle could be performed to ensure sufficient joint strength is achieved through later heating cycles such as a paint oven bake to meet the product service requirements. As such, cure cycles could be reduced to the minimal required to save manufacturing time and cost. Should the joint requirements change, a rapid model re-application would be able to identify the change in cure cycle required to meet the specification, avoiding repeated mechanical testing and extensive DSC evaluation.