Use of Bayesian inference in Analytical rheology

Transcription

1 Use of Bayesian inference in Analytical rheology Michael Ericson Scientific Computing, Florida State University Abstract There exists a simulator called rheology-bob2.3 that can predict linear rheology based on the structures, components and environment of molecules using tube field modeling 1. However, the inverse problem, finding the structure and components of a molecule based on linear rheology has not been solved. This inverse problem is nonlinear and therefore complex to solve. By using Bayesian inference, a simulator called rheology-bob2.3, and MCMC methods, we were able to create a program with some success that could figure out, the weight fraction, molecular mass, and structural archetype, given a choice between a linear polymer or a star polymer with three branches of a polyethylene melt. 1

2 Introduction High density polyethylene (HDPE) is linear and a strong type of polyethylene able to withstand corrosion and high temperatures. It readily crystallizes and is commonly recycled. It can be used for wood plastic composites ( fake lumber), molded goods, containing pyrotechnics (chemicals that produce heat and explosives), and a variety of other uses. By adding branches to polyethylene, it becomes much easier to process. It can move faster through pipes, and has many other properties that make it easy to process. However, the more branches added to polyethylene, the weaker it becomes. So we are left with an optimization problem: find the least amount of branches one can add to make it easier to process while retaining the same properties of high density polyethylene. The processing technology for making polyethylene is so advanced, that one can systematically choose which branches to remove from it 2. However, the problem is that when only a small amount of long-chain branches (~1 branch per 10,000 backbone atoms) are added, one cannot figure out the structure of these molecules 3. Analytic techniques like IR spectroscopy and chromatography fail to detect these long-chain branches due to their infrequencies. However, linear rheological properties are highly sensitive to very small changes in branching. By using a rheometer, which applies a vibration shear rate, we can obtain information on the linear rheology. Then based on methods that well be explained, we can find the structure and components in a polyethylene melt. Methods In Bayesian inference, Prob(H D) is the probability of obtaining the parameters, weight fraction, molecular mass, and structural archetype, given a choice between a linear polymer or a star polymer with three branches, 2

3 for a polyethylene melt based on the elastic and viscous modulus. We can ignore Prob(H) since for our current stage of research, we assumed no prior information of the parameters. Prob(D H) was found by comparing the elastic and viscous modulus calculated by bobrheology and the elastic and viscous modulus that would have been. The actual formula for comparison is below: error=.5*( (Σ(G'(w) d G'(w) h ) 2 ) + Σ(G''(w) d G''(w) h ) 2 ))) G'(w) d = Elastic modulus obtained from rheometer G'(w) h = Elastic modulus obtained by running the parameters in bob2.3-rheology G''(w) d = Viscous modulus obtained from rheometer G''(w) h = Viscous modulus obtained by running the parameters in bob2.3-rheology The smaller the error, the higher Prob(D H). We can also ignore for now P(D) since it only acts as normalization constant. We can therefore write the probability as Prob(D H) Prob(H D). 3 Our program sampled from many different values for the parameters since we wanted to obtain a probability distribution function, and there were many values the parameters could be. However, it would be computational expensive to calculate all possibilities of the parameters, so we used MCMC methods, specifically the Metropolis- Hasting algorithm to obtain a probability distribution 4. We sampled the data 10,000 times. Our value for figuring out the molecular weight was as followed: rho = 1.2 betam = rhom.^(2*rand(1,n) 1) Mp = M*betaM The reason we multiplied the molecular weight by a random number was that if the molecular weight was a large value, it would take too long for the program to reach it simply by adding a random number each time. Due to this Q(x',x)/Q(x,x') was equal to betam, since the probability of a random number being accepted was inversely proportional to its size (smaller random beta values had a higher

4 probability of being accepted). When creating a function to switch a polymer's structural configuration, the molecular weight had to be switched as well as the polymer's structural type, otherwise the move would most likely be rejected, since their viscoelastic properties are too different. Therefore we created a table of the star sand linear structures, and figured out which structure archetypes had the closest viscoelastic properties, and what the new value of the molecular weight would have to be of either a star or linear structure, in order to make a switch between a linear and a star polymer archetype with the closet viscoelastic properties. Results There were some simulations that were successful and others that failed. This first example is a successful run (shown below) The X axis is hard to read, but M1 show a probability density distribution in the range between 44,000 and 58,000. M2 shows a 4

5 probability density distribution between the range of 16,000 and 24,000. The results also show that there was a 100% chance that the polymer species in M1 was a linear polymer, while the polymer species in M2. The actual configuration and structures was that Species 1 was linear, with a molecular weight of 20,000, and 75% mass. Species 2 was also linear with a molecular weight of 90,000, and 25% mass. The true structure and components of polyethylene were the same as the program's results. The two species were a linear polymer and a star polymer, in which the linear polymer had a molecular weight of 50,000 and mass However, these plots show that the molecular weight for Species 2 is nowhere near 90,000. The graph displays a molecular mass density probability distribution region between 7000 and 20,000 amu. fraction.7, while the star polymer had a molecular weight of 20,000 with a mass fraction of 0.3. However, the simulation did fail a few times. Here is an example of one : 5

6 Based on data analysis, the results stated that a star-star blend had 96% likelihood while a linear-linear blend had a 0% likelihood. The reason for this error was that the program made a switch over to a star-star blend but did not switch again to a linear-star blend or to its correct configuration, linear-linear blend. This demonstrated that the most likely cause of error was in switching between polymer structural archetypes. The graph below compares the value of G' and G'' with the visoelastic properties of the program's resulting structure and configuration, and the viscoelastic properties of the actual structures and configuration. Once can see that the two viscoelastic properties are very different. Changes made The following two changes were made in order to improve the algorithm. 1. Using C++ instead of Octave reduced the amount of time it took the program to run. It went from taking from 6-8 hours to run, down to 2-3 hours. 2. Changing the algorithm for moving the species Instead of the algorithm used previously, a new algorithm was created. A pre-simulator used the MCMC Metropolis Hasting for a linear-linear, linear-star, and star-star configuration. However, the sample was much smaller, with only 1000 samples used. The lower configuration error was multiplied by gamma, (value = 1.2) and the parameters in the pre-simulator that were lower or equal to this number were stored during the simulation. When attempting to switch a species, the program chooses a random pre-simulated parameter and compared its likelihood with the current parameters. Although this method is better than the previous, it has some problems. Below are bar graphs generated from a run. Dr. Sachin Shanbhag rewrote the program from Octave to C++. This significantly 6

7 This might look like there are two likely possibilities for each species to be, but it s not. The two species actual made a switch between themselves. One species favors a molecular the other with 30,000 amu, one could easily figure out that the MW parameters favor the latter. Below are the three graphs, in which one can clearly tell the best fit. mass of 30,000 amu and the other favors a molecular mass of 80,000. Even so, just by running three sample: one with a molecular weight of both species with 30,000 amu, both with 80,000 amu, or one with 80,000 amu and 7

8 An even bigger problem with this method though is that it takes too long to run. It is very similar to just running the program three times, with one staying at a linear-linear configuration, another star-star, and the last one star-linear, which is not ideal. If there could be three or four types of polymer species, the program will take even longer. Conclusion The program is far from perfect and a few changes need to be made. One algorithm that needs to be created is a method to figure out if there are more polymer species, or less polymer species then two. While it is relatively easy to remove a polymer species, it is difficult to create a new species, especially when choosing what the species molecular weight and percent weight should be. Changes need to be made from changing the structural archetype of a polymer. Our assumption that the polymer with the closest viscoelastic properties is the most likely polymer to be accepted is not a true statement. 8

9 Instead, a method needs to be introduced to figure out the molecular weight of a star or linear structural archetype that would most likely be accepted. This does not necessarily have to be pre-assembled, but rather the program can learn what the best We can also add in prior information in the program, so it will be less likely for there to be more than one possible different structures and components. While gas chromatography Acknowledgments I would like to thank Sachin Shanbhag for coming up with an approach for this inverse problem and writing a lot of the code. I would also like to thank the National Science Foundation for providing funding for the project, and Florida State University for using their facilities. and light scattering techniques are inaccurate, these analytical techniques can at least provide us with some prior information. 9

10 Appendix Glossary Bayesian Inference: Prob(H D) = (Prob(D H)*Prob(H))/(Prob(D)). P(H D) : The posterior probability. It is the probability that the hypothesis is correct based on the data. P(D H): The conditional probability. It is the likelihood of obtaining that data if the hypothesis is correct. P(H): The prior probability The likelihood of the hypothesis before introducing the new data. Also known as the P(D): The marginal probability. The possibilities of obtaining the data based on both the hypothesis and other possibilities. P(D) can also be written as Ʃ(Prob(H 0 )P( H 0 )+ Prob(H l )P( H l )). Monte Carlo Markov Chains: Monte Carlo Markov Chains also abbreviated as MCMC methods are algorithms created probability distribution that are multivariate (many unknown variables) and stochastic (cannot be determined by simple means). The integral the MCMC creates is a probability distribution. As the name implies the algorithm uses systems based on Monte Carlo methods and Markov Chain methods. Markov Chain: Markov Chains are based on movement from one state to another. Each state is memory less, depending only based on its previous state. It is usually stated that X t+1 = X. In the case of Monte Carlo Markov Chains, the changes in parameters only depend on the previous defined parameters. Monte Carlo: The movement from one state to another is essential random. Monte Carlo Markov Chains can be thought of a drunken man walking. His position is space is based on where he moved last and the random movement he will make. 10

11 Metropolis Hasting In Metropolis-Hating, it starts off with randomly defined parameters. A random move is made, which is based on the previous move and the move is either accepted or reject. The move is based on the following two criteria: 1) Does the move have a higher or lower likelihood compared to the previous move? This can be written as P(x )/P(x) P(x ) = random move P(x) = previous move 2) Is the probability of entering the new state as returning back to its original state, without considering the likelihood of the move? For example if you are to multiple the original moves by a random number between 0 and 4 with equal probability, returning back to the original state is less likely. Therefore that must be taken into consideration. This can be written as: Q(x,x)/Q(x,x ) Q(x,x) = likelihood of entering the state based on the previous state Q(x,x ) = likelihood of returning to the state based on the new state Alpha = [P(x )/P(x)]* [Q(x,x)/Q(x,x )] 11

12 If the value of alpha is less than one then the move might be accepted. Even if the move is less likely than the previous, it is still important because: 1) It allows for one to escape a state. This will allow the walk to move to other states 2) A probability distribution is obtained rather than the highest probability The move will be accepted if alpha is greater than a random number between 0 and 1 with a uniform distribution. Polyethylene: Polyethylene is a polymer of magnitude greater than 1000 atomic mass units and contains a long hydrocarbon chain. Below is a picture of typical hydrocarbon chain: Picture from 5 The white balls represent hydrogen and the grey balls represent carbon. Rheology: Rheology is the study of flow and deformation of materials. The models used to describe behavior for materials in rheology are much more complex than those used in basic chemistry and constitutive Newtonian physics. Below are lists four examples of basic models used in basic physics and chemistry, and the difference between that model and the model used in rheology. A substance is a liquid, solid or gas. Liquids deform due to any amount of stress on in it, including 12

13 gravitational force and atmospheric pressure. Solids however do not deform readily under stress. When a solid deforms it will retain it shape once the stress is removed. Many polymers and complex mixtures cannot be classified as either a liquid or a solid. Instead, these materials are described as Viscoelastic. Viscoelastic materials have two components: the elastic modulus and the viscous modulus. The higher the value of the elastic modulus the material will retain solid-like properties. Likewise, the higher the viscous modulus, the material will display more liquid-like properties. The strain (deformation) on a solid is directly proportional to the stress. This is expressed in Hooke's law below: Stress = Strain * Elastic Modulus This formula does not accurately predict strain based on stress even for true solids. This formula can only be applied for relatively small amounts of stress. Larger amounts of stress will result in plasticity, in which a deformation may not be recovered, and the amount of strain based on stress is a nonlinear function. Newton describes the force on a fluid in one dimension as: Stress = (Velocity/(distance between two plates)) * (Viscosity) In other words, in order to double the velocity of a fluid one would need to double the force acting on it. Shearing, otherwise known as perpendicular force (aka friction), from the borders of the fluid act as an opposing force, which is why a fluid cannot accelerate under stress. Viscosity is not a constant. The viscosity can increase or decrease based on the shear rate, Viscosity decreasing as a result of an increase in shear rate is known as shear thinning. Viscosity decreasing as a result of an increased shear rate is known as shear thickening, the opposite effect of shear thinning. The amount of strain in a material is independent of time. Materials in rheology have what is known as a relaxation modulus, the amount of strain in a material changes based on how long a stress has been applied. 13

14 Viscoelasticity: G'(w) known as the Elastic Modulus, and G''(w) also known as the Viscosity Modulus. These two moduli are the linear rheological properties that were studied in order to figure out the structure and components of polyethylene. The Elastic Modulus is the measure of material strength measured as Stress/Strain. It is also a measure of the deformation that can be restored when the stress is removed. The Viscosity modulus is the resistance to flow, and can also be thought as deformation that is permanent. Shear stress is the force per area applied parallel to the surface. For example if you were to glide your hand across water, this would be an example of shearing. Shear rate on the other hand is the difference of shearing between layers of molecules. Molecules closer to the shear stress will have a higher level of shearing then those farther away. 14

15 Bibliography 1 Das, Chinmay, Nathaneal J. Inkson, Daniel J. Read, Mark A. Kelmanson, and Tom McLeish. "Computational linear rheology of general branch-on-branch polymers." Rheology 50.2, (2006). 2 MacKenzie, P. B.; Moody, L. S.; Killian, C. M.; Ponasik, J. A.; McDevitt, J. P. WO Patent Application , Spet. 17, 1998 to Eastman, priority date Feb 24, Baker, Clive, and Williams F. Maddam. "Infrared spectroscopic studies on polyethylene, 1. The measurement of low levels of chain branching." Die Makromolekulare Chemie (1993): Print. 4 Fox, Colin, and Antionetta Mira. "Geoff K Nicholls." Kuopio. June-July 04. Lecture. 5 Korkin, Anatoli, and Anastasia Alexandrovia. "Atomic Scael Design Symmetry." Atomic Scale Design Network. Web. 30 July