主導動力學曲線與三種不同動力學模型之比較研究

Comparison Study of Master Kinetics Curve and Three Kinetic models

王紹宇(Shao-Yu Wang)

指導教授 ： 鄧茂華

國立臺灣大學/理學院/地質科學研究所/碩士(2011年)

https://doi.org/10.6342/NTU.2011.00841

摘要

主導動力學曲線為一根據主導燒結曲線的概念，由化學反應速率方程式進一步推導而成之動力學模型。主導動力學曲線與其他動力學模型最大的不同之處在於不需要求得許多複雜的動力學參數，並且其預測主要是奠基於大量的實驗數據之上，因此相較於其他的動力學模型，主導動力學曲線有方便使用、預測準確、應用範圍廣泛等等諸多優點。前人研究中已經證實可以使用在諸如結晶動力學、熱分解反應、燒結緻密化速率等等領域，並且獲得不錯的成果。然而在使用的過程中，往往可以發現部分實驗資料雖然能夠得到良好的預測曲線，但其預測結果卻與由其他動力學模型所得出的預測相去甚遠。或者對於部分數據會發生擬合不良的情況，卻無法得知問題的確切原因。因此，了解主導動力學曲線與其它動力學模型在預測上的差異，以及釐清數據無法以主導動力學曲線擬合的原因，對於改進主導動力學模型是必要的工作。
為了瞭解主導動力學曲線與其它動力學模型的差異，本研究選擇了主導燒結曲線、Ozawa熱重分析法與Avrami方程式等三種動力學模型做為比較的對象。主導燒結曲線為一可完整描述燒結歷程的燒結模型，目前在科學界，已經有相當多與材料相關的研究使用主導燒結曲線來進行分析。Ozawa熱重分析法則改善了傳統熱重分析研究中，動力學參數不容易求得的問題，在許多熱重分析或化學反應的分析中都能夠提供準確的預測。Avrami方程式為研究晶體相變的動力學模型，可以使用簡單的作圖法來求得於持溫狀態下相變反應的反應機制值，以及反應程度與時間的關係線，在相變研究中扮演非常重要的角色。由於動力學模型的實驗數據得來不易，既無法大量獲得各種條件下的數據，也無法完全排除掉實驗誤差的因素，因此本研究使用合成數據來比較這些模型的差異，並配合前人研究的成果以及實驗資料互相比對，找出造成差異的原因。
由主導動力學曲線與主導燒結曲線的比較結果發現，具相同擬合概念的兩個模型，在預測能力上幾乎一致。這是由於大部分擬合資料中，溫度之間的差距都遠小於一個數量級，因此主導燒結曲線的log(Σ)函數多出來的1/T項並不會造成擬合曲線外型有明顯的改變，也不會影響到反應預測的結果。
由主導動力學曲線與Ozawa熱重分析法的比較結果中，可以發現主導動力學曲線可以良好地擬合零級反應以及一級反應的數據，而對於二級反應與三級反應的數據則會有「反應初期的資料點無法擬合於曲線上」的情況。這個現象可以由反應速率的觀點來解釋。由於零級反應與一級反應在反應初期速率較緩慢，符合S曲線於反應初期與後期的反應速率較慢的特性，而二級反應與三級反應在反應初期的反應速率最快，這點與S曲線本身的特色並不符合，因此無法利用S曲線得到良好的擬合曲線。這個結果說明了主導動力學曲線雖不能單純以S曲線來對所有反應進行分析，但透過不同外型的曲線還是有可能分析這些不同反應的反應數據。
主導動力學曲線與Avrami方程式的比較結果則說明了兩個模型僅會在反應中期處有一致的預測。而若以反應初期與反應後期的溫度去反推反應時間，則會發現反應時間有過慢或者過快的現象。數據的誤差，尤其是反應程度的誤差，可能導致這些不合理結果的原因。由於S曲線的特性，使得反應程度的誤差對於曲線兩端所造成的偏差遠大於對於曲線中段造成的偏差。因此，這些偏差導致主導動力學曲線對於低溫與高溫條件下的預測產生了嚴重錯誤的結果。
為了釐清部分數據無法以主導動力學曲線擬合的原因，本研究對於數據中的反應程度、溫度以及時間，分別加入±1%至±5%的隨機誤差，再利用主導動力學模型擬合，來瞭解這些誤差對於曲線擬合的影響。由結果可以發現反應程度(%)的對於主導動力學曲線的分析有最顯著的影響。這可能是因為對於主導動力學曲線的擬合而言，反應程度是直接影響到y軸擬合的參數，而時間與溫度則需先積分為log(Σ)函數才影響到x軸的擬合，因此反應程度的誤差對於擬合的影響最為明顯。
此外本研究亦使用不同組數的實驗數據來進行擬合，試圖找出能夠得出合理擬合結果的最小數據組數，由結果發現僅需要三組數據就能夠提供足夠合理的擬合結果。這顯示主導動力學曲線很能夠掌握數據的趨勢，能夠僅以少量的數據來預測出結果。因此確保輸入數據的正確性與準確性是非常重要的。
總結本研究的成果，由於主導動力學曲線的分析不需要先求得動力學參數，並且可透過不同的曲線對於不同反應進行分析，因此主導動力學曲線有非常大的潛力能夠成為具有最廣泛的應用範圍以及最方便使用的動力學模型。

關鍵字

主導動力學曲線 ； Ozawa熱重分析法 ； Avrami方程式 ； 動力學模型 ； 化學動力學

並列摘要

Master kinetics curve (MKC) is a new kinetic model derived from general chemical reaction rate equations and has only been studied for several years. The major difference between the MKC and other kinetic models is that it needs no assumptions regarding kinetic parameters. Instead, the establishment of the MKC is based on real experimental data. Thus, the MKC has several advantages, such as convenient to use, accurate to predict and a variety of potential applications. Previous researches have proved that MKC is capable of studying the kinetics of crystallization, thermal decomposition and sintering. However, some of the researches also showed that the predictions of the MKC, and in particular the parameter “apparent activation energy”, are not identical with the values determined by other kinetic methods. Therefore, it is necessary to study and understand these differences to make MKC into a universal kinetic model.
To compare the differences between the MKC and other kinetic models, three different kinetic models were chosen. The three models are the master sintering model (MSC), the Ozawa method and the Avrami equation. MSC is a well-established sintering model in the sintering field; it describes the whole densification process of a sintering. The Ozawa method is a kinetic model which greatly simplifies the determination procedure of kinetic parameters from thermogravimetric research. The Avrami equation is a phase transformation kinetic model which describes the relationship between reaction time and the extent of phase transformation; it is capable of determining the mechanism. In order to acquire enough data and reduce experimental errors, synthetic data and real data from early research are used for comparisons of the different models in this research.
The results showed that MKC and MSC have similar behavior in regard to prediction. The term 1/T of MSC introduces small deviations between these two models. If the reaction temperature increases, the deviations become slightly larger; however, these differences do not significantly change the predicted results.
The comparison between the MKC and the Ozawa method showed that the data from the 0th order and 1st order reactions can be perfectly merged by MKC’s fitting. However, the 2nd order and 3rd order reactions can only be partially merged by MKC analysis. These results can be explained by the reaction rates: 0th order and 1st order reactions have a slower reaction rate at the initial stage of reaction, and they are the same as an S-curve’s behavior (the S-curve has a slow reaction rate at the initial and final stages of reaction). However, the 2nd and 3rd order reactions have the highest reaction rate at the initial stage of reaction; they obviously differ from the characteristic of an S-curve. Consequently, the MKC cannot analyze all types of reactions by only using an S-shaped curve. Nevertheless, it is possible to merge these reaction data by other shaped curves.
The comparison between the MKC and the Avrami equation showed that the data only merged in the middle section of reactions. Predictions made by the MKC at both low and high temperatures are impossible. Data errors, especially the errors of reaction extent, might be the reason for the erroneous predictions. Due to the characteristic of an S-curve, errors of reaction extent might cause significant deviations to both ends of the curve rather than to the middle section. Likewise, these deviations give poor predictions at both low and high temperatures.
In this research, 1% to 5% errors were randomly added to three different variables of a MKC, i.e., time, temperature and reaction extent, in order to determine the relative influence of the variables on the models. From the results, we found that reaction extent affects MKC the most because the reaction extent is a parameter directly affecting the data fitting at the y axis. In comparison, time and temperature have to be integrated into the function log(Σ) at the x axis to affect the fitting. Therefore, the reaction extent requires greater accuracy than the other two variables.
In addition, we also used a different number of sets of data for the analysis of MKC in order to determine the minimum sets of data required to obtain a reasonable result. It shows that MKC can obtain a reasonable fitting curve by only three sets of data. This result also implies that, regardless of the accuracy of the data, MKC has the capability to describe the general trend of data. Thus it is important to ensure the accuracy of the data.
In conclusion, since MKC does not have to determine kinetic parameters and can analyze different reactions by different shaped curves, MKC definitely has the potential to become a universal kinetic model with a variety of applications and still remain convenient to use.

並列關鍵字

master kinetics curve ； MKC ； Ozawa method ； Avrami equation ； kinetic models ； chemical kinetics

參考文獻

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