(A! \97 .71 4.1.1: A 91 . I‘ltI‘ l v $311.5» 9:... llllllll lllllllllIllllllllllllllllll 3 1293 01109 3774 This is to certify that the thesis entitled Diffusion and Reaction of Small Molecules in Thin Polymer Films presented by Janice Lisa Tardiff has been accepted towards fulfillment of the requirements for M.S. degnmin Chemical Engineering axfifil‘btclfi L Major professor Date July 30, 1993 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution PLACE ll RETURN 30X to remove this chockout from your record. To AVOID FINEB mum on or baton duo duo. DATE DUE DATE DUE DATE DUE ____fi.__ ‘ “12,11 gill _ ”‘5‘”‘B‘L'mm m C ‘t 1 I, DE :1 We) DIPFDSION AND REACTION OF SMALL MOLECULES IN THIN POLYMER FILMS by Janice Lisa Tardiff A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1993 ‘ h a“ ma ABSTRACT DIFFUSION AND REACTION OF SMALL MOLECULES IN THIN POLYMER FILMS BY Janice Lisa Tardiff Diffusion and reaction in thin polymer films was studied for two different systems. The diffusion coefficient and kinetic rate constant for the sulfonation of polystyrene was estimated using the weight gain in the film during sulfonation, as measured by the extension of a quartz spring. Auger and FTIR analysis provided additional information about sulfonation, such as the required post- treatment of sulfonated material and the reaction mechanism in films such as polypropylene. In an epoxy/diamine system the diffusion coefficient of the diamine was estimated by the diffusion of a similar material, m-xylene, as determined by the weight gain in epoxy using an electrobalance. The kinetic rate constants were then estimated from extent of reaction data. The determined diffusion coefficients and rate constants provided the information required to model both systems. Results of the mathematical model were compared to experimental data to determine the accuracy of the model. \ tu- 0‘. Pa‘ Co; hel the n— Vt- hi: ACKNOWLEDGEMENTS There are many people who I wish to thank for both their help and support in completing this project. Firstly, I would like to thank my advisor, Dr. Eric Grulke, for his patience and helpful advice. I would also like to thank Coalition Technologies Ltd., in particular Bill Walles, for helping the sulfonation graduate students with the technolo- gy and for being more than willing to answer questions about the process. I must say a very big thank you to the staff at the Composite Materials and Structures Center. Dan Hook spent an incredible amount of time working with me to get the Auger analysis technique to work on sulfonated material, and I am extremely grateful. Mike Rich was very helpful in teaching me different analysis techniques and was very easy to work with. Brian Rook always had helpful advice. They are truly a professional staff. Several people provided other support which I am grate- ful for. I thank Dr. Alec Scranton for many good conversa- tions and suggestions. Rod Andrews was a tremendous help in the lab and it is doubtful that this project would have been completed at this time without his aid. Adam Havey was instrumental in the design of the sulfonation apparatus. I must also thank my parents, Richard and Florence Koehler, for their love and support. They have never al- lowed me to say "I can't" and have been behind me in every endeavor that I have undertaken. Finally, I thank my husband Joe for his love, patience, and support. He has had to deal with me through the ups and downs of this project and is probably even more relieved than I about its completion. But he has been wonderful through it all and I must sincerely thank him. TABLE OF CONTENTS List of Tables List of Figures I. II. III. IV. Introduction Gas Reacting with Polymeric Surfaces A. Theory 1. Fluorination 2. Chlorination 3. Sulfonation General Mathematical Model 1. Theory 2. 'Model for Sulfonation Experimental Methods 1. Experimental Setup 2. Results of Experiments 3. Determination of Diffusion Coefficients and Rate Constants 4. Further Studies Analysis of Sulfonated Material 1. Auger Analysis a. Theory b. Sample Preparation c. Method Development 2. FT-IR Analysis a. Theory b. Experimental Results Liquid Polymerization in a Thin Surface Film A. Analysis of an Epoxy-Diamine System B. Mechanisms for the Epoxy/Diamine Coupling C. Equations Describing the System D. Determination of the Solubility of Diamine E. Determination of the Diffusion Coefficient F. Determination of the Kinetic Rate Constant Appendices A. Appendix A - Basic Program for Modelling Diffusion B. Appendix B - Basic Program to Model Sulfonation C. Appendix C - Weight Gain from Sulfonation D. Appendix D — Basic Program to Model Curing of an Epoxy References 131 132 134 138 141 146 W Table Table Table Table Table List of Tables Dimensions of samples for sulfonation 38 Results of experiments with kinetic apparatus 48 Experimetally determined diffusivity and rate constant 58 Experimental solubility of xylene in epoxy 109 Comparison of Least Squares Error for Constant and Exponentially Dependent Diffusivities 114 Pi Fig Fig Fig Fig Fig Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Table of Figures Grid for solving partial differential equation explicitly. Concentration of diffusing species vs. time at a position of 1 micron from the interface. Concentration of diffusing species vs. time at a position of 1 micron from the interface. Concentration of diffusing species vs. time at a position 15 microns from the interface. Concentration of diffusing species vs. time at a position of 1 micron from the interface. Concentration of diffusing species vs. time at a position of 1 micron from the interface. Schematic of the sulfonation reaction Schematic of quartz spring experiment Mass uptake in polystyrene vs. time for run #1 Mass uptake in for run #1 polypropylene vs. time Mass uptake in for run #2 polystyrene vs. time Mass uptake in for run #2 polypropylene vs. time Mass uptake in for run #3 polystyrene vs. time Mass uptake in for run #3 polypropylene vs. time Mass uptake in for run #4 polystyrene vs. time vi 18 22 23 24 25 26 28 33 39 4O 41 42 43 44 45 Fig Fig Fig Fig Fig Fig rig Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Mass uptake in polypropylene vs. time for run #4 Comparison of model to experimental results for the sulfonation of polystyrene in run #1 Comparison of model to experimental results for the sulfonation of polystyrene in run #2 Comparison of model to experimental results for the sulfonation of polystyrene in run #4 Comparison of model to Auger line scan Result of interaction of electron beam with surface electrons in Auger analysis Schematic of how sample block is to be trimmed for Auger analysis Low magnification of sulfonated polystyrene embedded in epoxy High magnification of sulfonated polystyrene embedded in epoxy Sulfonated, unneutralized HDPE embedded in epoxy Sulfonated, unneutralized PP embedded in epoxy Sulfonated, unneutralized PS embedded in epoxy Low magnification of sulfonated, unneutralized PP embedded in acrylic High magnification of sulfonated, unneutralized PP embedded in acrylic Low magnification of sulfonated, unneutralized PS embedded in acrylic High magnification of sulfonated, unneutralized PS embedded in acrylic vii 46 55 56 57 60 63 67 71 71 74 74 75 78 78 79 79 H. Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Low magnification of sulfonated, neutralized PP embedded in acrylic High magnification of sulfonated, neutralized PP embedded in acrylic Low magnification of sulfonated, neutralized HDPE embedded in acrylic High magnification of sulfonated, neutralized HDPE embedded in acrylic Sulfonated, neutralized PS embedded in acrylic Comparison of IR spectrums of unsulfonated and polypropylene Comparison of IR spectrums of unsulfonated and sulfonated HDPE Comparison of IR spectrums of unsulfonated and sulfonated polystyrene Schematics of epoxy system Reactants in epoxy system Reactions occurring during the cure of the epoxy Structure of m-xylene Schematic of electrobalance equipment Weight gain with time for run #1 using the electrobalance Weight gain with time for run #2 using the electrobalance Weight gain with time for run #3 using the electrobalance Comparison of experimental results to model results for run #1 Comparison of experimental results to model results for run #3 viii 81 81 82 82 83 sulfonated 89 90 92 94 96 97 105 106 110 111 112 116 117 Figure Figure Figure Figure Figure Figure Experimental extent of reaction Comparison of model to experimental extent of reaction Determination of k0 and Ea Predicted concentration gradient of primary amine after two hours at 75' C Predicted concentration gradient of secondary amine after two hours at 75' C Predicted concentration gradient of tertiary amine after two hours at 75' C ix 123 125 126 128 129 130 Introduc'_ The films is literatur carbon (11 is not We in isolat Process. many ind reaction attempt and reac and the Chapter 1 Introduction The diffusion of small molecules into thin polymer films is a process which is well characterized in the literature. Examples include the diffusion of oxygen or carbon dioxide into polymer films. Diffusion with reaction is not well characterized, however, due to the difficulties in isolating the diffusion process from the reaction process. This is an interesting phenomenon since there are many industrial processes which depend on the diffusion and reaction of small molecules within polymer films. An attempt is made in this thesis to characterize two diffusion and reaction problems; the sulfonation of thin polymer films and the cure of an epoxy/diamine system. Sulfonation is a technique which modifies the surface of polymeric films using sulfur trioxide vapor. The vapor diffuses into the films and reacts to provide sulfonic acid groups within a thin surface layer of the film. This thin layer is polar and reduces the permeation of many nonpolar molecules through the polymer. For example, sulfonation has .been used on the inside of plastic fuel tanks to reduce the emission of organic vapors into the atmosphere. The 'technique may also be used to recycle plastics by making clifferent plastics miscible through the interaction of the polar groups on the surface of the films. changing ' Use does not allows ti Which re: Welles, in the u helped t Howeverl been We? Provide In 2 Sulfonation can also be followed by metallization, which replaces the hydrogen atom of the sulfonic acid group with a metal ion. This provides the polymer film with completely different surface properties. Potentially, the barrier can be modified to provide specific properties by changing the counter-ion of the sulfonic acid group. Use of sulfonation to modify the surface properties does not alter the properties of the bulk polymer. It allows the polymer to be used in particular applications which requires it to have specific barrier properties. Bill Walles, of Coalition Technologies Ltd., has been a pioneer in the use of sulfur trioxide vapor for sulfonation. He has helped to promote the use of this technology in industry. However, the diffusion and reaction of sulfonation has not been well characterized. It is hoped that this thesis will provide a better understanding of the sulfonation process. In the epoxy/diamine system, a fiber is coated with the epoxy, which is then exposed to a stoichiometric mixture of epoxy and the diamine. The diamine diffuses into the epoxy coating and eventually cures the coating. Ideally, the «epoxy coating around the fiber is cured to the fiber .interface, to provide protection and strength to the fiber. The determine the extent of cure of the epoxy, the diffusion cnoefficient and rate constants for the curing reactions must be characterized. A technology such as this has many important applications. It is hoped that a model of this system w the cure coating. Bot concentra temperatu determine technique measure t balance, cathEtone t0 the d but the which pr thESQ 5y 3 system will provide enough information to help to optimize the cure cycle and provide complete cure of the epoxy coating. Both systems will be characterized in terms of their concentration- dependent diffusion coefficients and their temperature-dependent rate constants. Methods used to determine these variables include an electrobalance technique, using a similar, but non-reactive species, to measure the diffusion coefficient, and a quartz spring balance, which measures the weight gain with time using a cathetometer. These techniques only provide approximations to the diffusion coefficients and reaction rate constants, but the approximations allow the systems to be modeled, which provides a better understanding of what occurs within these systems. 6&8 Reac Theory A c film is has beer chlorina chlorin. the mai the met t0 fluC fluorir SUlfong litera1 furthe: Plu°ri2 F has be surfac tmms. Chapter 2 Gas Reaction with a Polymeric Surface Theory A common method of modifying the surface of a polymeric film is to expose the film to a reactive gas. This method has been used in processes such as fluorination, chlorination, and sulfonation, which use fluorine gas, chlorine gas, and sulfur trioxide, respectively. Although the main interest of this thesis is the sulfonation process, the methods used to model this process can also be applied to fluorination and chlorination. Present knowledge of fluorination and chlorination can also be used to understand sulfonation. Therefore, it is important to review the literature of these surface modification techniques before further research is conducted. Fluorination Fluorination is a surface modification technique which has been used to improve the barrier properties of plastic surfaces. It has been shown that fluorinated plastic fuel tanks on automobiles reduce the emissions of volatile «organics from the fuel system, as compared to the emissions :Erom untreated plastic fuel tanks. The barrier improvement is due to the addition of fluorine radicals on the polymer backbone, which increases the polarity of the polymer. and has Flu a mixtur diffuses fluorine PIOpertie material , microns . times . bECauSe . almost n The with the backbone the Car}: amount c oxifluor ‘[~CHz where F* 5 These polar groups repel nonpolar organic compounds in the fuel. The fluorination process has been well characterized and has found a significant amount of use in industry.1 Fluorination occurs by exposure of polymer surfaces to a mixture of fluorine gas and an inert gas. The mixture diffuses into a thin layer of the polymer surface and the fluorine reacts with the surface to provide modified surface properties. To modify the barrier properties of a polymeric material, only a thin surface layer, on the order of 10 microns, must be treated. This requires short exposure times. Longer exposure times are generally ineffective because the improvement in barrier with longer times is almost negligible.2 The fluorine gas initiates a free radical reaction2 with the polymer to abstract a hydrogen from the carbon backbone of the polymer or from a substituent located off the carbon backbone. It has been determined that if a small amount of oxygen is present during fluorination, oxifluorination occurs. 02 ’E‘CH2C32'1n‘ + F2 ----- > -[-CH2CHOF-]n- + F* Eqn. 2.1 where F* is the fluorine radical. The oxyfluorinated Surface provides better barrier properties than the fluorinated surface and is the preferred treatment. It is usually very difficult to prohibit the presence of oxygen in tune system, particularly in a manufacturing atmosphere, as 6 it is likely to be present as an impurity in the gas, as residue in the reactor, or absorbed in the polymer.3 Surface property modifications as a result of fluorination include a lower dispersion energy, a lower polar energy, and a higher hydrogen-bonding energy. A lower surface energy is expected since fluorocarbons have lower surface energies than hydrocarbons, but the increase in hydrogen-bonding energy is unexpected. The change in the hydrogen-bonding energy is most likely due to the presence of oxygen in the modified surface as a result of oxifluorination. Hydrocarbon solvents are retained within the fluorinated surface even though they may wet the surface because of these property modifications. Aromatic solvents, however, are not well retained because they are more polar. Fluorination of plastic fuel tanks and bottles is usually conducted during the blow-molding process. Fluorine gas, or another fluorine containing gas, is introduced during the expansion stage of the blow-molding operation at a recommended concentration from 0.1-20% by volume in the gas. The precise fluorine concentration depends on the desired surface modifications and the material being used, but for polyethylene, a concentration of 3.5% by weight is ‘usually sufficient.4 Chlorina- Chlorina taxmiqu lkmid, tmum.vi< limitati 96$ usua particui Problem light 0 Centers numbe r indust‘ W been i Chlorf “Ptak. 1:1th Th£a f Chlorination Chlorination is another polymer surface modification technique that uses chlorine gas or a chlorine containing liquid, such as sulphuryl chloride. The technique has not found widespread application, however, due to its many limitations. Exposure of the polymer surface to chlorine gas usually does not result in a high level of chlorination, particularly for crystalline materials.5 To overcome this problem the polymer surface must treated using ultraviolet light or electron beam irradiation to produce free radical centers before exposure to chlorine gas. This presents a number of problems when the technique is used on an industrial scale. When chlorine gas is exposed to surfaces which have not been irradiated or exposed to ultraviolet light, some chlorination has been found to occur in polyethylene. The uptake of chlorine is fast in the first 5-10 minutes and much slower for further exposure times up to 120 minutes. The fast chlorine uptake is attributed to reaction in the amorphous regions of polyethylene, while the slow uptake is attributed to reaction in the crystalline regions. In the amorphous region the density of the polymer is much less since there is not an ordered crystal lattice structure as there is in the crystalline region. Therefore, the chlorine gas can diffuse more quickly through the amorphous region.6 Since the chlorination reaction occurs quickly, the greatest limitat region into th To sample i irradiat. which res POlymer. some of t radical c Chlorinai Where c1 I:eaCtiOn Sites an irradiat lar98r c larGEr a decrease Chlorine 510w t0 Of the i haZardOu EchOds 8 limitation to the chlorination reaction in the amorphous region is the extent to which the chlorine gas can diffuse into the sample. To enlarge the amorphous region of the polymer, the sample is treated by ultraviolet light or electron beam irradiation. These techniques produce free radical centers, which results in damage to the crystal structure of the polymer. The change in the crystal structure will affect some of the properties of the polymer. Once the free radical centers are produced, the propagation of chlorination occurs via; -C*H- + C12 ------ > -CC1H- + Cl* Eqn. 2.2 -CH2- + c1* ------- > -C*H- + HCl Eqn. 2.3 where Cl* is the chlorine radical. Termination of the reaction occurs when the number of available chlorination sites approaches zero.6 As expected, higher doses of irradiation result in a greater uptake of chlorine due to a larger crystalline area which is damaged and, hence, a larger amorphous region. The level of chlorination decreases with decreasing temperature due to the lower chlorine gas diffusivity at lower temperatures. A major problem with the technique is that it is too slow to be of practical value. Another problem is that both of the irradiation and ultraviolet light methods are hazardous due to the production of hydrochloric acid. The methods are also impractical for use in industry due to the difficu. ultravi Ch; also rec polymer azobisis chlorine The radi backbone The chai Unfortup this rea and envj USQ. sulfona. Mu< soluuoI swellin, method ‘ gas holy the bill: EffiCie: however 9 difficulty in exposing the entire surface area to ultraviolet light or the electron beam. Chlorination using a liquid such as sulphuryl chloride also requires the production of a free radical center on the polymer chain.7 In this case an initiator, such as azobisisobutyronitrile (ABIN), is used to abstract a chlorine atom to produce the sulphuryl chloride radical. The radical then abstracts a hydrogen atom from the polymer backbone to produce a radical site on the polymer chain. The chain is then chlorinated using sulphuryl chloride. Unfortunately, hydrochloric acid is also a side product in this reaction, as well as sulfur dioxide, which are health and environment problems. Therefore, the use of this technique for chlorination has also not found widespread use . Sulfonation Much work has been done on solution sulfonation, but solution sulfonation causes problems such as polymer swelling and solvent contamination. A gaseous sulfonation method is preferred, using sulfur trioxide gas. The small gas molecule can diffuse farther into a polymer sample than the bulky liquid group, resulting is a more thorough and efficient surface modification. Sulfur trioxide gas is, however, difficult to use due to its high affinity for water. gaseous The which me oxygen a doublxa‘b °Xqun a and time with a Where POlyme sulfut Protg Since f0unc 8033C the ‘ and Wate 10 water. Much of the work that has been done to characterize gaseous sulfonation has been conducted by Walles. The sulfur trioxide molecule is a resonance hybrid, which means that the oxygen atoms are equivalent. The oxygen atoms are strongly bound, with a large degree of double bond character. Due to the binding nature of the oxygen atoms, the sulfur atom is strongly electron-deficient and the oxygen atoms are electron-rich.8 A typical reaction with a polymer is: -[CH2CH2]n-CH2CH2- + $03 ----- > -[cnzcnz]3-cnzcnso3u- Eqn. 2.4 where the sulfonic acid group (-SO3H) is added to the polymer chain. Therefore, in reactions with polymers, the sulfur atom attacks the electron-rich sites, abstracts a proton, and the oxygen atoms accept the acidic protons. Since sulfur trioxide often exists as a complex, it has been found that the modified surface group may initially exist as $03803H, with one 803 group being more strongly bound than the other. With a water wash after sulfonation, however, one 803 group is removed as H2804, thus producing the SO3H group on the polymer chain.9 The sulfonation reaction has a small negative enthalpy and is a reversible reaction. Due to its strong affinity for water, the sulfonic acid group may be removed with water, producing sulfuric acid.10 To prevent the loss of 'hn.neut complex due to and the T. unneut Mflyme the rn a QCK: 3991i tum. ions Cal] 11 the sulfonic acid group, the sulfonated material is neutralized with ammonia or ammonium hydroxide: -[CH2CH2]n-CH2CHSO3H- + NH3 ----> -[CH2CH2]n-CH2CHSO3'NH4+- Eqn. 2.5 The neutralized sulfonic acid group (so31nyfi) is stable complex which resists the attack of solvents such as water due to the strong attraction between the sulfonic acid ion and the ammonium ion. The neutralized sulfonic acid group, as well as the unneutralized sulfonic acid group, is polar, thus making the polymer water wettable. Water-wettability is important for the recycling of plastic materials. The group also provides a good barrier to nonpolar solvents and gases for use in applications such as sulfonated plastic fuel tanks. To further change the surface properties of plastics, metal ions can be exchanged for the ammonium ion in a process called reductive metallization.11 Reports by Ihata12 indicate that during sulfonation a polymer radical is produced. The polymer radical may then react with the sulfonic acid group to produce the sulfonated polymer, or the elimination of a hydrogen atom may occur to form an unsaturated bond within the chain, with the formation of sulfurous acid. _ i The form accompai yellow- unsatur and po; This 1 Sulfur argumg Appii publ; mate? the Sulf of p et. Poly impr( Coupe inter and t 12 -CH2-CH2-CH2- + so3 ---> -CH2-CH2-CH°- + 50311 ---> -CH2-CH2-CHSO3H- or ---> -CH2-CH=CH- + H2803 Eqn. 2.6 The formation of the unsaturated bond within the chain is accompanied by a color change for clear materials to yellow-brown and then brown. This side reaction to form the unsaturated bond occurs in materials such as polyethylene and polypropylene, but not in materials such as polystyrene. This indicates that the side reaction only occurs when sulfur trioxide attacks the polymer backbone. A stronger argument for this will be given later in this thesis. Applications of Sulfonated Polymers Several studies of sulfonated polymers have been published which indicate the usefulness of sulfonated material due to the modified surface properties. Many of the studies to be cited use material which has been sulfonated using solution methods, but they show the types of potential applications for sulfonation materials. Wang et. al.13 have used sulfonated polystyrene in blends with poly(ethyl acrylate 4 vinyl pyridine) (P(EA-co-VP)) to improve the compatibility of the materials. The enhanced compatibility of the materials is due to the ionic interactions between the negatively charged sulfonate groups and the protonated nitrogen atoms on the P(EA-co-VP) chains. Hara et. exhibite whether solvent: attract enhance polar s The p0] ion cor enhanc. were e Verifi W betweE found °f Sui LiSPs. EOre ‘ Since temps imprg and p 13 Hara et. al.14 showed that sulfonated polystyrene ionomers exhibited different solution properties depending upon whether polar or nonpolar solvents were used. In nonpolar solvents the ionomers tended to form aggregates because of attraction between the ion pairs. The aggregation was enhanced by increasing the ion content of the ionomer. In polar solvents the ionomers showed polyelectrolyte behavior. The polyelectrolyte behavior was enhanced with increasing ion content. Therefore, the intramolecular attraction was enhanced by nonpolar solvents and the intermolecular forces were enhanced by polar solvents. Hara's results are verified by the results of MacKnight and Lundberg.15 Weiss et. a1.16 studied the improvement in miscibility between sulfonated polystyrene and Nylon 6 blends. They found that Nylon 6 was completely miscible with metal salts of sulfonated polystyrene, with miscibility improving by LiSPSo.5. For this comparison, D=5.5x10'18 m2/sec, At = 30 sec, and Ax=1x10'8 m. The results for this case are shown in Figures 3.3a and b. It was found that at long times the calculated results diverge from the error function values and continue to grow larger for longer times. This appears to be a concern, but the level of divergence is small enough to be negligible. At 100 minutes, it was found that the difference between the model values and the error function values were less than one percent. Therefore, because of the good correlation between the model and the error function, the model was expanded to include the reaction. Model for Sulfonation The model of sulfonation was somewhat more complicated that the above model due to the fact that the diffusion coefficient of sulfur trioxide was dependent upon concentration and the chemical reaction was included. There are three species whose concentrations changed during the reaction: sulfur trioxide ($03), unsulfonated polymer (UP), and sulfonated polymer (SP). Figure 3.4 is a schematic of the sulfonation reaction. The model had to include finite difference methods to calculate the changing concentration 28 —% 0:3.ch — 0+4.ch }— + $03 —-> CH3 CH3“ IH ‘SgosH +CH20 - CHzCH J— l - _l CH3 CH3“ "[LCHZCH - CH2CHj— + 803 —> L A n 1- Cr-izpr-l - CngCr-I ‘Jr 803H I; '7’ Sulfonated Region I Bulk Polymer 303 + N2 Film sore-5 Figure 3.4 - Schematic of the sulfonation reaction 29 of each of the species. Assuming that the sulfonation reaction is a first order reaction, the rate equations for each of the species are: fit 6x 503 6x _kcsoscup Eqn. 3.11 500P_ 6 D 50w ‘ UP 6t 6X 6X 'kCSOSCUP Eqn. 3.12 5Csr=_ 5 D 503p ‘ SP 6t 5x 6x .,. kCSQCUP Eqn. 3.13 However, since the unsulfonated polymer and sulfonated polymer have diffusion coefficients which are approximately zero, due to the large size of these species, the last two rate equations reduce to: 60 ”Pp/(08030!” Eqn. 3.14 6t 568;: irhe rate equation for each species was written in the grid 30 form of the Saul’yev method, as described in the opening paragraphs. The initial conditions for the system are as follows; At t = o; c503 = c°503 = 0 CUP = C UP CSP = COSP = 0 In other words, at time zero, there is no sulfur trioxide gas present in the system and therefore, no sulfonated polymer. The concentration of the unsulfonated polymer is the mass of the polymer divided by its molecular weight and its volume. Boundary conditions are not required for the unsulfonated and sulfonated polymer because there is no coordinate dependency on its concentration, but are required for sulfur trioxide, the diffusing species. The $03 boundary conditions are; Atx=OCSo3=C1 Atx=°0 Cso3=C2=o ‘ ‘ The second boundary condition is a result of the semi-infinite media assumption in the system. In the first boundary condition, C1 is equal to the solubility of the diffusing species (in this case 303) in the polymer. In the previous example, C1 was assumed to be 3 mol/L. The actual value of the solubility of $03 in any particular polymer is difficult to determine and was one of the objectives of this work. If the 803 vapor does not condense on the surface of 31 the film, the $03 solubility should be the concentration of $03 in the gas phase. Since there is no data in the literature on the diffusion coefficient of 803 in polymers or on the rate constants for the reaction, these parameters must be experimentally determined before the system can be modelled. The following section describes the method for determining these parameters. The results of the mathematical model are then discussed. Chapter 4 Experimental Methods for Sulfonation Experimental Setup To obtain experimental data for the determination of the diffusion coefficient of sulfur trioxide and rate constant for sulfonation, a quartz spring balance and cathetometer technique was used. Although other techniques, such as the use of electrobalance, may have been preferable to obtain precise data, use of such a method was impossible due to the corrosive nature of the reactant. A diagram of the experimental system is shown in Figure 4.1. As shown in Figure 4.1, the polymeric film was suspended in the sample chamber using a quartz spring. An important step before beginning the experiment was to ensure that the chamber was tightly sealed. This was important not only for the safety of the researchers, but to prevent a leak in the system which would cause the vapor to condense on the film and result in false data. As the polymeric film was exposed to sulfur trioxide vapor, the quartz spring extended, and the extension was measured using a cathetometer. Recordings of the extension using the cathetometer were somewhat crude because of the flow rate of vapor through the sample chamber caused the sample to bounce and resulted in errors in the extension measurement. 32 33 5 ON: J. Ill, tomcopcoo It So ONI T | Ft T E mom IN /\ dob. o3m>oEom canmso BnEmm BaEmm actnm Ntmsd It So mom Figure 4.1 - Schematic of quartz spring experiment 34 Since the cathetometer measured the extension of the quartz spring, in centimeters, the weight gain in the sample due to sulfonation was determined using the spring constant. The spring constant, with units of milligram per millimeter, was multiplied by the spring extension, resulting in mass gained. The source of the $03 was an $03 generator obtained from Coalition Technologies Ltd.23 The $03 generator consisted of a reactor containing oleum which was internally circulated with a liquid pump when not in use. During use, a gas pump was used to externally circulate the gas through the experimental system. The experimental system, in this case, consisted of a kinetic test chamber in the form of a condenser, as shown in Figure 4.1. The sample chamber was made of glass to allow the viewing of the quartz spring and polymer sample. Water in the condenser portion of the sample chamber was kept at a constant temperature using a refrigerated water bath. Rates of reaction are usually highly dependent upon temperature. Therefore, in an effort to isolate the diffusion process from the reaction, low temperature experiments were used. These low temperature experiments were used to estimate the diffusion coefficient, without reaction, of 803 in the polymeric films. An assumption was made that only diffusion occurred in these experiments, although, in reality, some reaction did occur. Once the 35 diffusion coefficient was determined, higher temperature experiments, where both diffusion and reaction occurred, were used to identify the reaction rate constant. To prepare for an experiment, the chamber was tightly sealed and purged with nitrogen gas to remove any volatiles that might have been present in the system. Exit gases from this purge went through a gas scrubber containing sodium hydroxide to neutralize any acid vapors before being released to the atmosphere. The valves on the inlet and exit lines were then closed and a vacuum was pulled on the system. This step was important for the removal of any condensed liquids on the film, sides of the sample chamber, or in the stainless steel lines of the system. Failure to pull a vacuum on the system would result in condensation of liquid on the film surface and an exaggerated weight gain. After the vacuum step, the pressure of the system was increased to atmospheric pressure using nitrogen gas. In the meantime, the gas pump was turned on and the $03 generator was left on internal circulation until that system equilibrated, as set by a timer built into the system. To begin an experiment, a measurement of the polymer sample height was made using the cathetometer before 803 was introduced into the sample chamber. The 803 generator was then switched to external circulation and readings of the extension of the quartz spring were taken using the 36 cathetometer every 30 seconds. Typical experimental times were ten minutes. An attempt was made to determine the vapor concentration in the sample chamber by taking a vapor sample, mixing it with water, and then titrating the liquid solution. This technique was not, however, reproducible and therefore not reliable. Problems occurred with use of this technique, such as leakage of vapor from the syringe, leakage of vapor from the sample port, and condensation of water vapor or residual acid within the syringe which gave false measurements. Thus, to determine the concentration of $03 at the sample interface, or the boundary condition of the problem, the solubility was determined using the weight gain data from the quartz spring and cathetometer experiment. In the experiment, the mass uptake with time was determined. To determine the concentration of $03 at the interface, the mass uptake was divided by the molecular weight of $03 to determine the number of moles within the polymer film. The volume of the polymer film was determined by multiplying the thickness of the sulfonated layer within the film by two times the surface area of the film. The two is included in the equation since two faces of the film are exposed to the vapor. Figure 4.9 shows that the thickness of the sulfonated layer is approximately six microns. An in-depth discussion of this figre is postponed until latter 37 in this thesis. Thus, the concentration at the gas-solid interface was the males of 803 divided by the volume of polymer. An assumption was made that the volume change of the sample with sulfonation was negligible. This method was used for all experimental conditions, since the concentration of $03 at the interface changed with gas phase concentration and reaction temperature. Results of Experiments Two polymer films, polystyrene and polypropylene, were sulfonated using the quartz spring balance technique. The polystyrene samples had dimensions 2 cm x 4 cm and the polypropylene samples were about 28 mm x 40 mm. The thickness of the samples were determined using a microscope and a micron scale. A sample of each polymer was run at four different sulfonating conditions. A summary of the dimensions and weight of the samples is given in Table 4.1. The samples were prepared for sulfonation by washing with ethanol to remove any surface contaminants. A 1/16 inch hole was punched in each sample to allow room for the hook on the quartz spring. A sample was then loaded into the sample chamber and the run began as described above. Figures 4.2a and b show the results of run #1, which is the weight gain in polystyrene and polypropylene, respectively, with time. In run #1, the reactor temperature was 90' F (32.2‘ C) and the temperature in the sample 38 chamber was 11' C. Figures 4.3a and b show the results of run #2, which again the weight gain of PS and PP with time. Run #2 differs from run #1 only in the chamber temperature of 20' C during the experiment. Figures 4.4a and b show the results of run #3. In run #3, the reactor temperature was 110' F (43.3' C) and the chamber temperature was 11' C. Figures 4.5a and b show the results of run #4, which differs from run #3 only in the chamber temperature, which is 20' C in this case. Different reactor temperatures resulted in different vapor phase concentrations of 803. At higher reactor temperatures, it was expected that the concentration of $03 in the vapor phase was higher. Sample Length Width Thickness Volume (mm) (mm) (microns) (L x 107) (g) Mass Run #1- 40 20 54.26 4.3408 0.0463 PS Run #1- 40 26 53.62 5.5765 0.0501 PP Run #2- 40 20 54.26 4.3408 0.0464 PS I Run #2- 40 27.5 53.62 5.8982 0.0499 40 20 54.26 4.3408 0.0456 53.62 6.0054 0.0510 54.26 4.3408 0.0445 6.0054 39 .8305: own 00h 0mm coo 0mm 00m one 00v 0mm oon 0mm CON em? 00— On 0 C’TITITIIITIJTi’T’TIITII 0 800.0 «000.0 - «000.0 I @0000 wooed 0000.0 I 5000.0 800.0 I I I 800.0 I 50.0 I I 200.0 «80.0 I I 200.0 300.0 900.0 .2 9.89.. «no... Figure 4.2a - Mass uptake in polystyrene vs. time for run #1 40 . .3306: com 0mm 00m .0mb 00x. 0mm 000 Can 00m One 000 0mm 00m 0mm 00m 09 00— 0m 0 n n n u “ u u n x u u u u w “ IIIIIIITIlJ 0 1. ~86 I . . 1 coed I -- mood I -r mood I i 5.0 I -- ~86 I .3. 8.39. mmoE I -- Sod . I 1: 0.5.0 I .- Sod . i 8.0 .. «Nod . -r .436 [I owed Figure 4.26 - Mass uptake in polypropylene vs. time for run #1 41 .03.. 2:: 0m¢ 00v 0mm 00m 0mm 00m 03 00— 0m 0 1 l i i l i i i 1 IrlIilIIII 1r. . I .- I I -4 I I I .- I I e I .. 500.0 N000.0 @0000 @0000 m000.0 0000.0 5000.0 a000.0 moood 900.0 500.0 N Sod . n 50.0 0 900.0 m 50.0 0 50.0 3. 9.3% 82: Figure 4.3a - Mass uptake in polystyrene vs. time for run #2 42 .2508: 86. one 68 6.: 62 63 86 63 666 one 8.. 68 con S~ oo~ 6m. 8. on 6 xiuiiuxiuivuuluilfllle I I .r ~86 I I i 30.0 866 J. 866 I .- 56 I : ~86 I I .i 30.0 3.3.86: 32: : 266 I -- 6.66 -- ~66 4 -66 L- «36 I .. e~66 I .- 686 I L- 86 Figure 4.3b - Mass uptake in polypropylene vs. time for run #2 43 68.960 8~— 8: 86. 68 68 8~ 86 68 8.. 68 8~ 8. 6 u n u “ “ n I - u n u I n J 0 41 2000.0 II I in «000.0 II II in 0000.0 II .1 0000.0 II - 6866 III L. 0000.0 I I. - 886 I - 6866 - 886 L- .86 - :86 I - ~_86 .- 986 - 386 I - £86 I 4 6.86 - :86 . - 386 I - 986 .0. 8.89.. was: Figure 4.4a - Mass uptake in polystyrene vs. time for run #3 44 000 000 00¢ .08. 2:0 1 l 1 I 1 I 1 I 1 I I I 1 i 1 1 1 1 l l 000.0 000.0 000.0 N 5.0 0 _.0.0 0 5.0 50.0 A«00.0 $4.00 00.0 000.0 000.0 000.0 N000 mvod 30.0 50.0 .2 9.86: nmaE Figure 4.4b - Mass uptake in polypropylene vs. time for run #3 45 .03. 2:: 8a.. 00 p F 000 F 000 000 005 000 000 00v 000 00w 00 p 0 w w w w w 1 1 w w u w n I 0 1- «000.0 - 6866 - 6866 I - 666 II - ~F86 I - 386 - $86 886 I - ~86 I - -86 I 1. 886.. I - e~86 - 6~86 - ~86 . .2 9.86: 368 Figure 4.5a - Mass uptake in polystyrene vs. time for run #4 46 38.0.5. 68 8... 8m 68 8.. 68 con om~ 8~ 8. 8. on 6 m n n n 1 w x u I 0 I - 866 I 1.. 000.0 -- 866 I - ~86 I -.. 6.66 I - 6.66 I - .86 I - 686 I .2 8.8g: 82: - ~86 I - 86 I - ~86 I - 686 I -.. 686 I - ~86 I . i 000.0 I .. - 866 - .86 _ p p - q q — Figure 4.5b - Mass uptake in polypropylene vs. time for run #4 47 A temperature of 11° C was the lower temperature limit of the water bath, since lower temperatures caused the sample chamber to frost, making sample viewing impossible. It was hoped that runs #1 and #3 would isolate the diffusion process fairly well from the reaction and that runs #2 and #4 would give an indication of the kinetics of the process. There was a concern about condensation on the polymer film sample during the sulfonation process. To determine if this was the case, a teflon sample was suspended from the quartz spring and the weight gain with time was measured. Teflon was used since 803 does not react with the material or diffuse into the material. Therefore, any weight gain on the teflon would be due only to condensation. Teflon samples with the dimensions of 6.75 mm x 5.64 mm, a thickness of 1.46 mm, and a weight of 0.1122 g were suspended in the sample chamber with the same quartz spring, which had a spring constant of 300 mg/450 mm. It was found that after ten minutes there was no weight gain on the teflon. Therefore, it was concluded that condensation on the polymer film was not a factor in this experiment. Using the weight gain of $03 in polystyrene for each sulfonation condition, the concentrations of $03 at the gas- solid interface were estimated for each case. It is noted that in Figure 4.2a, 4.3a, 4.4a, and 4.5a, the weight gain in polystyrene appears to have reached a steady state value. 48 Therefore, the concentration of 803 at the interface can be determined by the method described above with great confidence. These values are reported in Table 4.2. E: Run # Percent Mass Concentration of Uptake 803 at the interface (mol/L) 1 - PS 2.74% 1.6485 1 - PP 48.56% NA 2 - PS 3.30% 1.9946 2 - PP 58.81% NA 3 - PS 4.09% 2.4287 3 - PP 79.35% NA 4 - PS 6.44% 2.6022 4 - PP 85.81% NA I__— Table 4.2 - Results of experiments with kinetic apparatus In Figures 4.2b, 4.3b, 4.4b, and 4.5b, for the sulfonation of polypropylene, it is noted that at the end of the experimental run the weight gain within the polymer film has not reached steady state. The sulfonation reaction for polypropylene is much more complicated than that for polystyrene, due mainly to side Therefore, only crude estimates sulfonation of polypropylene in been nice to have continued the state conditions were obtained, reactions which occur. can be made about the this thesis. It would have experiment until steady but the weight gain due to sulfonation was beyond the limit of the quartz spring. Further measurements would have resulted in failure of the spring. A spring constant with a higher spring constant 49 could have been used, but would not have been sensitive to the weight gain at low times. Therefore, the concentration of 503 at the gas-solid interface for polypropylene could not be determined with any accuracy. An interesting observation in the literature was that some polymer samples changed color from transparent to dark brown during sulfonation.11 This observation was also made when polyethylene samples were sulfonated by this author. It is believed that this color change was due to a side reaction which occurred during sulfonation. This reaction is; -(CH2-CH2-CH2)n- + nSO3 ---> -(CH2-CH=CH)n- + nsto3 Eqn.4.1 The sulfurous acid produces the color change of the samples and such reactions has been reported by Ihata.12 The presence of the conjugated bond in the polymer backbone for sulfonated polyethylene and polypropylene has also been supported by FT-IR experiments, as reported in Chapter 6 of this thesis. With long reaction times, the color change will worsen and a layer of the sulfonated film will eventually slough off when rinsed with water. Therefore, a degradation of the polymer surface occurs when these samples are sulfonated. This is an important observation because the color change occurred within one to two minutes of sulfonation at low 803 concentrations for polyethylene and 50 polypropylene. These results are also supported by the high percent mass uptake in polypropylene reported in Table 4.2 and the linear relationship of mass uptake in polypropylene versus time shown in Figure 4.2b, 4.3b, 4.4b, and 4.5b. In run #4, the percent mass uptake for polypropylene was 85% after 8 minutes. It is unlikely that such a high mass uptake is due strictly to the diffusion of $03 and the reaction to form sulfonated polymer. It is more likely that the sulfurous acid formed by the side reaction condenses on the film to produce the large weight gain. Therefore, the reaction mechanism for sulfonation in polypropylene is much more complicated than originally expected. As noted in Table 4.2, the percent mass uptake for the polystyrene after twenty minutes was much smaller than that for polypropylene. There was also no color change for PS upon sulfonation. A possible explanation for this lack of color change is that $03 preferentially attacks at the para position of the benzene ring of PS.24 Only at very long sulfonation times is it possible that the 803 may attack the polymer backbone. Therefore, the formation of sulfurous acid in the sulfonation of PS is not important in normal exposure times and the side reaction does not need to be included in the mathematical model for polystyrene. 51 Determination of the Diffusion Coefficients and Rate Constants To determine the diffusion coefficients and rate constants for the sulfonation process, the mathematical model must predict the experimental curves of mass uptake versus time shown in Figures 4.2a, 4.3a, 4.4a, and 4.5a. An analysis of sulfonation in polypropylene will not be made since there the mechanism is more complicated than expected and therefore requires a more in-depth analysis to be completely understood. It is believed that the side reaction plays an important role in the sulfonation of polypropylene, but a precise determination of the reaction mechanism is beyond the scope of this thesis. Any determination of the diffusion coefficient and reaction rate constant for the sulfonation of polypropylene based upon the data presented above would certainly be false. To predict the experimental curves, Equations 3.11, 3.14, and 3.15 were solved using the mathematical model. As a reminder, these equations are; 5030,_ a C 50:- E n. 4.2 at 6x “3 6x ”“30”” q 5CRW 6! =7koquUP Eqn. 4.3 52 =*kcsa,CUP Eqn. 4.4 The sulfonated polymer (SP) was taken to mean polymer which has sulfonic acid groups present, while UP is the unsulfonated polymer. It should be noted that the diffusion coefficient for sulfur trioxide in equation 4.2 is differentiated with respect to x, which is the distance within the polymer slab. Inclusion of the diffusion coefficient within the derivative is necessary since it is dependent upon concentration. Crank notes that the concentration dependence of the diffusion coefficient of vapors in high-polymer substances is a marked, characteristic feature.19 A commonly used form of D is; D=Doexp‘c Eqn. 4.5 where Do is the diffusion coefficient in the limit of zero concentration and a is a variable determined empirically. This form of the diffusion coefficient is used in the model in this thesis, since it provides the best fit for the experimental data. Using the mathematical model developed in this thesis, a constant diffusion coefficient was first found which fit the curve in Figure 4.2a. Another constant diffusion 53 coefficient was found which fit the curve in Figure 4.3a. Values for Do and a were then chosen which provided diffusion coefficients similar to the constant diffusivities previously determined. The values for Do and a which best fit the experimental curves of Figures 4.2a and 4.3a were; 130 = 1.37 x 10'1“ mz/sec i 0.3 x 10'14 a = 0.41 i 0.09 Remembering that Figures 4.2a and 4.3a are the curves for runs #1 and #2, which were at different temperatures, Do also includes whatever temperature effects which may have been present. In other words, the form of Do is actually; - E. DO=Dmexp ”7 Eqn. 4.6 Although the sample chamber temperature was the same for run #3, the experimental data for run #3, as shown in Figure 4.4a, is not characteristic of the other polystyrene experiments. Since there is a question about the reliability of the data shown in Figure 4.4a, it was left out of this analysis. In the next step of the model, the kinetic rate constant was determined. In Table 4.2, the mass uptake in polystyrene for run #4 was twice as much as for runs #1 and #2 for the same experimental time, indicating that reaction probably played a greater role. Therefore, using the diffusion coefficient determined above, the kinetic rate 54 constant was determined by fitting the model results to Figure 4.5a. The kinetic rate constant was of the form; k=kaexp "E. Eqn. 4.7 I?T The kinetic activation energy was estimated to be 10 kcal/mol. Therefore, to determine the rate constant for the sulfonation reaction, ko was varied in the model until the theoretical results fit the experimental data. The resulting value for k.o was: k0 = 2.22 L/mol i 0.09 This value for kO provided reaction rate constants on the order of 7.7 x 10'8 L/mol. A comparison of the theoretical results to the experimental results for polystyrene runs 1, 2, and 4 are given in Figure 4.6, 4.7, and 4.8, respectively. It is noted that in all three figures the reaction is probably complete after about 300 seconds and there is little weight gain due to diffusion at longer times. To summarize, the determined parameters for the diffusion coefficient and the rate constant are; 55 I I 1 1 1 1 1 I I I I I 350 400 450 500 550 000 650 Tin. (no) 250 150 100 0.0015 T 00014 - 0.0013 - 0.0012 -* 0.0011 -. 0.001 - 0.0000 Tr 0.0000 - 0.0007 -. 0.0000 ‘- 0.0005 ~t 0.0003 - 0.0002 0.0001 1 1 Figure 4.6 - Comparison of model to experimental results for the sulfonation of polystyrene in run #1 56 00v 00v Gib 000 ‘- db db .8... r... G. 00—. 300.0 «000.0 0000.0 300.0 0000.0 0000.0 508.0 0000.0 0000.0 30.0 p 30.0 N .000 0 30.0 . 300.0 0 50.0 300.0 (Biennium for the sulfonation of polystyrene in run #2 Figure 4.7 - Comparison of model to experimental results 57 com. _ 00—. - -l* - .68. 35.. 68. 68 68 8. 86 8... ....T-...-...._ -- . . ..-.-.--.i_-..-i- _. 00v Figure 4.8 - Comparison of model to experimental results for the sulfonation of polystyrene in run #4 g Do a ko 1.37 x 10'14 0.41 i 2.22 L/mol mz/sec 0.09 i 0.09 i 0.3 x 10'1“ T E2=I.e 4.3 -E§perimentally determined diffus and rate constant v ty To further test the accuracy of the mathematical model, the model was used to predict the results of an Auger analysis line scan across the cross-section of the polystyrene film. The Auger line scan produced a concentration profile for sulfur into the film, indicating the thickness of the sulfonated layer. When the model was used with the parameters determined above, it was found that the predicted thickness of the sulfonated region in the film was greater than that shown by the Auger line scan. However, the sulfonated polystyrene had been rinsed and neutralized with ammonium hydroxide before the Auger analysis. As noted previously'o, studies had indicated that the sulfur trioxide vapor sulfonated the film in the form of a dimer (SO3SO3H). Sulfonation by means of the dimer had been observed in the weight gains studies with the quartz spring and cathetometer. Thus, the diffusion coefficient and rate constant reported above are based upon sulfonation by the dimer. However, with water washing and neutralization, it has been indicated that the second 803 group is removed, leaving only the sulfonic acid group on the polymer. Therefore, the amount of sulfur trioxide 59 observed in the Auger line scan is most likely about half the amount observed in the quartz spring studies. Thus, the mathematical model must be modified to provide a fair comparison to the Auger line scan by halving the concentration at the vapor-solid interface. In other words, the boundary condition was modified. The film analyzed by the Auger line scan had been sulfonated under the conditions of run #2. To predict the results of the line scan with the model, the concentration at the interface, or the boundary condition at x = 0, was reduced to half the value reported in Table 4.2. Thus, the boundary condition was; At x = 0, Csny== 0.9973 mol/L This alteration provided a good (it to the Auger line scan data. A comparison with the Auger line scan is given in Figure 4.9. Since the model results fit well with the Auger line scan, it can be concluded that the diffusion and reaction mechanism for the sulfonation of polystyrene has been well characterized. Further studies Although the concentration dependent diffusion coefficient and the kinetic rate constant have been estimated for polystyrene in this thesis, it is believed by this author that further studies are required to provide a better understanding of the sulfonation process. FUN-HIHJH Concentration 60 ms I IN! I’C 12/18/92 LLZSI RD) 1 LINE 1 8CD 11545.36 HIN. llll: 95213 Sulfonated ps / liOf SCflE "KIN: 0.2fikc/s. WISH: 0.ch/S “=10!“ 01:0.W ii i d ,HI 1. ii 1' i '1 44!- ‘4 ‘. 1 ~ i 4| . - l ‘- - I 31' Willi. I'l- I"| i i ii' i :ll 111:1 iii! I‘ll“ Ii ' i P "z ' - 'é £1.14 - ill-ii ii iii-Iii “- ‘i'i'i‘ “Mini"! l'ml ”iii 1 - ' "l :"v, i 4 l|"'1| I'iiiii‘i' ii I i i i” ' Ilii'lii'iliIiI‘J s . ‘ ll; 1'4" 1 l ““1 414’1’41 4111' H 7 4 DISINKI. I'ISICI‘OOS 8 10 Concentration profile Position (microns) Figure 4.9 - Comparison of model to Auger line scan 61 Sulfonation was found to be much more complex than originally believed, particularly for polypropylene and HDPE. A more controlled experimental setup is required for sulfonation of those materials, and a more in-depth kinetic study is needed to understand the reaction mechanisms. In particular, the effect of neutralization on the reaction rate expression still needs to be characterized. It is also believed that the desorption studies are necessary to determine the amount of $03 which remains reacted in the polymer. Use of a vacuum would provide a convenient desorption method in a reasonable time frame. Desorption is important since it will remove the second 803 group of the dimer and show the actual weight gain within the film after rinsing and neutralization. Chapter 5 Analysis of Sulfonated Materials Auger Analysis Theory: Auger electron spectroscopy (AES) is a surface analytical technique which uses an electron beam to excite the sample. Electrons from the electron beam interact with electrons on the surface of the sample being analyzed and force the formation of the electronically excited electron A* by a transfer of energy to the sample. The mechanism for this process is; A + ei' ---> A* + e’i' + ea' Eqn. 5.1 where ei‘ is the incident electron from the source, e’i‘ is the same electron after it has interacted with A and lost energy, and ea' is the electron which has been ejected from an inner orbital of A.24 When an electron is ejected from an inner shell, an electron from an outer shell fills the resultant vacancy. Figure 5.1 shows a schematic of this process. The Auger electron, ea', has a kinetic energy Ek which is independent of the energy of the electron which originally created the vacancy in the inner orbital of A. The kinetic energy of the Auger electron is the difference 62 63 i E" V Valence EV electrons EV -_.-.-__.____--_. — -- E"b . . Core ED 9 f? i electrons E b CI; Figure 5.1 - Result of interaction of electron beam with surface electrons in Auger analysis 64 between the energy released in the relaxation of the excited electron (Eb - Eb’) and the energy required to remove the second electron from its orbital (Eb').24 Therefore; Ek = (1::b - Eb’) - (Eb’) = Eb - 2Eb’ Eqn. 5.2 The electron beam is scattered upon entering the sample and will either back-scatter to the surface or proceed to the inner orbital of the sample. After incident electrons reach the inner orbital of the sample, the resulting Auger electrons lose energy as they travel through the solid by inelastic collisions with bound electrons. However, if Auger electrons are released close to the surface, they can escape with little energy loss and be detected by an electron spectrometer. An Auger spectrum, therefore, produces information on back-scattered electrons and Auger electrons. Output from the spectrometer produce peaks due to the Auger electrons which are superimposed on a background of back-scattered electrons. It is important to note that the Auger electrons are very sensitive to the state of the surface and to its topography.25 Therefore, the sample surface must be as smooth as possible and a careful sample preparation technique is necessary. The exciting electron beam voltage is chosen to provide an adequate excitation of the Auger electrons and to produce a desired spatial resolution. Usually the beam voltage is 65 chosen between 5 and 10 eV. However, in this analysis the beam voltage was lowered to reduce charging effects. Charging is a problem when insulating materials, such as polymers, are analyzed. It results when the total current reaching the specimen is greater than that leaving through back-scattering and Auger electron emission. To overcome charging problems, the sample can be rotated so that the incident beam is at a glancing incidence and the primary beam energy can be reduced. Both of these methods where used in this analysis. The primary components of the Auger electron spectroscopy system are the electron gun, an electron spectrometer, an electron detector, and an ion gun. The ion gun can be used for specimen cleaning and depth profiling. During specimen cleaning, the ion gun is typically an argon beam which erodes a small area of the specimen. Sample preparation: Cross-sections of polymer films which were sulfonated for up to 10 minutes in a gas mixture of up to 10% by volume 303 in nitrogen where analyzed by Auger analysis to determine the concentration gradient of S03 through the film. Since these films were quite thin, on the order of 50 microns thick, an extensive preparation technique was 66 necessary before the analysis could be conducted. This technique was developed by Kalantarz6 for analysis of the fiber-matrix interphase in polymer composites and is a modification of the standard Transmission Electron Microscopy microtoming technique.27 Sulfonated films were first embedded in an epoxy matrix, which was allowed to cure. The type of epoxy used in the analysis was critical and will be discussed later. There must be enough epoxy around the film to allow the sample block to be trimmed with a razor blade. The epoxy around the sample was trimmed at an angle so that a trapezoid containing the film and a small amount of epoxy stands above the rest of the sample block. An example of how the sample was trimmed is shown in Figure 5.2. Caution must be taken when trimming the sample block so that the adhesion between the epoxy and the polymer film is not disturbed because a disturbance results in a loss of adhesion between the materials and cause a gap between the film and the epoxy. This prevents trimming the epoxy very close to the film. The purpose of the Auger analysis is to scan the film for sulfur from the side which has been sulfonated to the center of the film. The result is a concentration profile for sulfur across a cross-section of the sulfonated film. If a gap between the epoxy and the 67 polymer film epoxy Figure 5.2 - Schematic of how sample block is to be trimmed for Auger analysis 68 film appears, the sample cannot be analyzed due to charging problems in the area of the gap, which distorts the image on the sulfonated side of the film. The gap distorts the smoothness of the surface and probably causes Auger electrons and backscattered electrons to be released in the gap in various directions, rather than towards the spectrometer. The sample block was then microtomed using a glass knife to provide a smooth surface. Microtoming will probably be necessary before the sample block is trimmed with the razor blade so that the embedded polymer sample can be easily identified. The purpose of the microtoming procedure is not to section the sample, but to provide a smooth surface for analysis by the electron beam. Once a reasonably smooth surface has been obtained, the sample block is removed from the microtome specimen grip and placed in a gold plasma coater, with the microtomed side facing up. A thick gold coating (approximately 100 nm) is placed upon the sample block to alleviate charging on the sample surface. The gold coated sample block is then returned to the microtome for a final trimming by the glass knife to remove gold from the surface being analyzed and to provide a smooth surface. The surface being analyzed was, however, surrounded by a region of gold, which reduced the problem of charging. 69 Method Development: After preparing the sample as described above, the sample was placed on the sample stage in the spectrometer and exposed to a vacuum overnight. Pumping the sample for such a long period of time was necessary to remove volatiles from the sample, such as unreacted curing agent from the epoxy. Once the sample was ready for analysis, the beam voltage was set at 3 eV and the electron gun was turned on. It was immediately apparent on all of the samples which were analyzed that charging was still a problem even though the samples where sputtered with a thick coat of gold and the incident beam was rotated to a 30 degree glancing incidence. Therefore, it was necessary to sputter the sample with argon from the ion beam. This technique carbonized the sample surface, which means that carbon atoms where placed on the surface. These carbon atoms aided the transfer of energy so that the charging effect was significantly reduced. An attempt was made to skip this sputtering step by mixing carbon black with the epoxy as the sample was embedded, but it was found that this method had little effect on the charging problem. The main problem with sputtering the sample with argon was that not much else other than carbon and sulfur could be detected because Auger electrons for other elements did not possess enough energy to be detected. For example, it would 70 have been nice to have identified nitrogen to determine if all of the sulfur trioxide groups were with ammonium hydroxide. However, the energy of the nitrogen Auger electrons was so low compared to that of the back-scattered electrons that there was no indication that nitrogen was present. The technique also made it difficult to detect sulfur, but with persistence the sulfur peak was identified. Upon viewing the sample through the scanning electron microscope (SEM) portion of the Auger electron spectrometer, it was apparent where the bulk of the sulfonic acid groups lay within the sulfonated polymer film. A bright line along the interphase of the film and the epoxy matrix was seen, which contained sulfonic acid groups. This line was brighter than the rest of the sample block due in part to the higher electron density of this region, because of the sulfur and oxygen atoms. Figure 5.3 shows this effect at several magnifications using the scanning electron microscope. The material in the figure is sulfonated polystyrene, which had not been neutralized. Figure 5.3a is the sample at a lower magnification, with the bright region ‘between the unsulfonated film and the epoxy readily apparent. The epoxy is on the left side and the film on the right. Figure 5.3b is the sample at a higher magnification. The thin strip through the center of the sample is the sulfonated region. An interesting observation is this , "r. -‘P-. .x. \ L . o .“§.:;;z__“kc \{ ”epoxy, {fag . 1‘ 7'” ‘ ' , “_lr ‘ ”Signxnvfiur. rd; 52-” ' . 13 Figure 5.3a - Low magnification of sulfonated polystyrene embedded in epoxy. Epoxy on the left, film on the right. Figure 5.3b - High magnification of sulfonated polystyrene embedded in epoxy. 72 figure is that there appears to be two interfaces. One is between the epoxy and the sulfonated region of the film and the other is between the sulfonated region and the unsulfonated region of the film. Based upon the scale given in Figure 5.3b, the sulfonated region appears to be about 1.4 microns thick and is a uniform thickness through the sample. Before sulfonation the thickness of the polymer film was about 54 microns, indicating that only a very small layer of the polymer was sulfonated in approximately 10 minutes. Obviously, the surface properties of the sulfonated layer is much different than that of either the epoxy or the bulk polymer film. A point analysis was conducted with the Auger electron microscope within the bright region of the sample to ensure that sulfur was present. A test acquisition for sulfur was then made to determine the maximum energy of the sulfur Auger electrons compared to the energy of the back-scattered electrons. This test aquisition was done as precisely as possible to get the best signal-to-noise ratio. The sample was then magnified to the appropriate level, which depended upon the thickness of the polymer film. A line scan for sulfur was conducted to get the concentration gradient of sulfur trioxide through the sample. Figure 5.4a,b, and c show the results of the line scan for sulfur in sulfonated high density polyethylene, 73 polypropylene, and polystyrene. These samples were unneutralized and embedded in a commercial epoxy. In Figure 5.4a, the interface is at approximately the center of the photograph, with the epoxy on the left side and the polyethylene film on the right. The line scan for sulfur peaks at the interface, with sulfur found on each side of the interface. This result was quite different than the expected gradient which is at a maximum at the interface and decreases at distances toward the center of the film. The only practical explanation for why sulfur is observed in the epoxy is due to diffusion of sulfur trioxide out of the film and into the epoxy. Figure 5.4b is sulfonated polypropylene embedded in epoxy, with the polymer film on the left side and the epoxy on the right. The peak concentration of sulfur is at the interface of the epoxy and the film, but there is a gradient of sulfur into the epoxy, indicating diffusion of the sulfonic acid groups into the epoxy. Figure 5.4a is sulfonated polystyrene embedded in epoxy with the film on the left and the epoxy on the right. Again, the sulfonic acid groups appear to be diffusing out of the sample since the peak in the sulfur peak is at the interface of the film and the epoxy. These results where somewhat perplexing, so another commercial epoxy was used, but the same results were obtained. These results suggest . 4;: Figure 5.4a - Sulfonated, unneutralized HDPE embedded in epoxy. Epoxy on the left, HDPE film on the right. -15... 11% ‘ .interface . 1;, Figure 5.4b — Sulfonated, unneutralized PP in epoxy. Epoxy on the right, film on the left. .4c — Sulfonated, epoxy. Film on the left, epoxy on the right. interface epoxy unneutralized PS embedded in 76 that the sulfonic acid groups are not strongly bound to the polymer films, regardless of the type of polymer used. Upon investigation it was found that both of the commercial epoxies used were mercaptan based. Several hypotheses were then made concerning the diffusion of the sulfonic acid groups out of the polymer films. It is possible that the mercaptan groups in the epoxy were attracting the sulfonic acid groups. It is also possible that there was excess amine in the epoxy which attracted the sulfonic acid groups, in the same way as a neutralizing agent such as ammonium hydroxide readily reacts with the sulfonic acid groups. Both of these hypotheses indicated that the sulfonic acid groups are not strongly bound to the polymer, which is a concern if one considers the applications of the sulfonation technology. It was also suggested that the local temperature of the sample may have risen during the Auger analysis and maybe forced the reverse reaction to occur. This possibility is not likely to be the sole reason for the observations. First, not all of the samples showed beam damage, but all of them showed indications of sulfur trioxide diffusion into the epoxy. Therefore, the rise in the local temperature is not a solid reason for the observations. Secondly, for the reverse reaction to occur, there are indications that it is necessary for water to be present. Since the sample was exposed to a vacuum overnight - 77 to remove the volatiles, including water, it is unlikely that any water was present in the system. Therefore, a different explanation must be given for the diffusion of the sulfonic acid groups out of the polymer film. Unneutralized, sulfonated samples where next embedded in an acrylic based epoxy to see if the mercaptan groups in the commercial epoxy were causing the problem of the sulfonic acid groups diffusing out of the polymer film. Figure 5.5a, b, c and d show the results of the Auger analysis. Figure 5.5a is sulfonated polypropylene embedded in acrylic, with the entire polypropylene film running through the center of the photograph. Figure 5.5b is the same sample at a higher magnification with the sulfur line scan. The acrylic is on the left side and the film is on the right. Once again, the peak sulfur concentration is at the interface of the acrylic and the film, but there is a concentration gradient to the left of the peak, out of the film and into the acrylic. Figure 5.5a is sulfonated polystyrene embedded in the acrylic, with the polymer film on the left and the acrylic on the right. The line scan for sulfur is shown in Figure 5.5d. Again, the peak concentration is at the interface, but there is a concentration gradient of sulfur into the acrylic. In this case there are no mercaptan groups present to attract the sulfuric acid groups. Therefore, it appears that the Figure 5.5a - Low magnification of sulfonated, unneutralized PP embedded in acrylic. Figure 5.5b - High magnification of sulfonated, unneutralized PP in acrylic. Acrylic on the left, film on the right. 79 Figure 5.5c — Low magnification of sulfonated, unneutralized PS embedded in acrylic. Figure 5.5d — High magnification of sulfonated, unneutralized PS embedded in acrylic. Film on the left, acrylic on the right. 80 sulfonic acid groups are diffusing out of the film to react with excess amine groups within the epoxy. This is a strong indication that the sulfonic acid groups are not strongly bound to the polymer film. It is clear that the unneutralized sulfonic acid groups are not stable within the polymer film. The obvious question is whether neutralization of the sulfonic acid groups stabilizes these groups within the polymer film. Polystyrene, HDPE, and polypropylene were all sulfonated and then neutralized in ammonium hydroxide for approximately 24 hours. They were then embedded in the acrylic based epoxy and analyzed in the same manner as the other samples. The results of the line scans for sulfur for these samples are shown in Figure 5.6a, b, c, d, and e. In these cases, the bright area where sulfonation has taken place is still quite clear, but there is a distinct boundary between the epoxy and the polymer film. Figure 5.6a and b are sulfonated and neutralized polypropylene at low and high magnifications, respectively. In Figure 5.6b, with the acrylic on the left and the film on the right, all of the sulfur is contained within the sulfonated region of the film. There are no concentration gradients into the acrylic. However, for this sample, the peak sulfur concentration is not at the acrylic/film interface. Instead, the peak is halfway into the sulfonated region. 81 :igure 5. 6a - Low magnification of sulfonated, neutralized PP embedded in acrylic. 4;.-. “< s'y'oblil'iwu". ”I. '3 1‘ ‘1 Figure 5.6b - High magnification of sulfonated, neutralized PP in acrylic. Acrylic on the left, film on the right. Figure 5.6c - Low magnification of sulfonated, neutralized HDPE embedded in acrylic. ii; :1 HDPE embedded in acrylic. 83 interface: §11/@9/92 3.0kV 1@.okx Figure 5.6e - Sulfonated, neutralized PS embedded in acrylic. Acrylic on the left, film on the right. 84 The initial low concentration region from the interface to the peak concentration can be attributed to an oversulfonation of the film, resulting in the side reaction of carbon double bond formation in the backbone of the polymer. This side reaction was indicated by the change in color of the sample from clear to black. Even though double bond formation occurred, some sulfonic acid groups were still present since the side reaction had not gone to completion. Figure 5.6c and d show sulfonated and neutralized HDPE. The entire film, embedded in acrylic, is shown in Figure 5.6a. The very bright region on the right side of the film seems to be caused by a separation of the film from the acrylic since attempts at analyzing this region were futile due to charging problems. The interface on the left side appears to have two boundaries, which is the sulfonated region of the film. Figure 5.6d show the results of the line scan for sulfur in this region. Once again, all of the sulfur was contained within these two boundaries, with no indication of the sulfonic acid groups diffusing out of the sample into the acrylic. The fact that the peak sulfur concentration is not at the interface may be again due to the side reaction which was indicated by a slight color change in the sample. 85 Figure 5.6e shows sulfonated and neutralized polystyrene. The acrylic is on the left and the film is on the right. Again, all of the sulfonic acid groups are contained within the sulfonated region of the film. In this sample there were indications of beam damage on the sample. However, this beam damage did not result in sulfonic acid groups diffusing out of the sulfonated region, indicating that neutralization fixes the sulfonic acid groups within the film, at least for the short times of the Auger analysis. A reason for the shift in the peak sulfur concentration away from the interface is not understood at this time, since there was no indication of the side reaction by a color change. These results make clear the importance of neutralization to the sulfonation process. Without neutralization, the sulfonic acid groups can be attracted out of the sulfonated material and the sulfonated layer may eventually be lost. Possible neutralizing agents include ammonia gas and ammonium hydroxide. The question at this point is whether the bond between the:sulfonic acid group and the polymer film actually exists, and if so, how strong is it? According to the CRC Handbookga, the bond strengths of polyatomic molecules, the C-S bond in C6H5CH2-SCH3 is about 59.4 kcal/mol, while the 86 C-H bond in H-CH(CH3)CH2 is about 83 kcal/mol. C-H bonds for other molecules are listed, with the lowest bond strength being about 70 kcal/mol. Therefore, it is possible that sulfonic acid groups could be fairly easily removed after reaction and replaced by a hydrogen atom. It is also possible that the sulfonic acid groups may be trapped within the polymer film without reaction occurring and with neutralization the bulky salt group cannot escape from the film. It is believed by the author that there is a bond between the sulfonic acid group and the polymer film, but it is weak and neutralization is absolutely necessary to hold the sulfonic acid group within the film. It may be necessary for long term studies to be conducted to determine if the neutralized groups eventually diffuse out of the polymer film. FT-IR ANALYSIS Theory In infrared spectroscopy, a sample is exposed to infrared radiation, typically in the range of wavenumbers from 4000 to 400 cmfl. The energy corresponding to these wavenumbers (or wavelengths) is not enough to cause electronic transitions, as in Auger spectroscopy, but they can cause groups of atoms to vibrate with respect to the 87 bonds that connect them. Molecules absorb energy at specific wavelengths because the vibrational transitions of the groups of atoms correspond to distinct energies.10 For a molecular vibration to be observed in the infrared spectrum, it is necessary that the dipole moment of the bond under consideration be nonzero. A bond with a zero dipole moment does not interact with the electric field of the electromagnetic wave. Therefore, there is no change in the dipole moment and, hence, no absorption of energy.10 To determine the infrared spectrum, a sample beam passes through a sample cell and a reference been through the reference cell. A rotating mirror alternately allows light from each of the beams to enter the monochromator. The monochromator allows only one frequency of light to enter the detector at a time. The resulting signal from the detector indicates the difference between the intensity of light in the sample and reference beams. Experimental As noted previously in this thesis, it was observed that HDPE and polypropylene films turned brown upon sulfonation, while the polystyrene film remained colorless. It was suggested by this author, based upon findings by Ihata, that a side reaction was important HDPE and 88 polypropylene which resulted in conjugation in the backbone of the polymer and the release of sulfurous acid. To determine if this hypothesis was in fact true, infrared spectrums were determined for both unsulfonated and sulfonated films. Figure 5.7 compares the infrared spectrums for unsulfonated and sulfonated polypropylene. The sulfonated polypropylene sample had become dark brown through the reaction process. Sulfonated polypropylene has a much higher absorbance near 3200 cm'l, indicating the presence of the hydroxyl of the sulfonic acid group. What is more interesting, however, is the high absorbance around 1680 -1 cm. in the sulfonated polypropylene spectrum. In comparison, there is no absorbance at 1680 cm’1 in the unsulfonated spectrum. Absorbances at or near 1680 cm’1 usually indicative of a carbon double bond. Since the magnitude of the absorbance is large, these spectrum appears to give strong proof of the presence of the side reaction in the sulfonation of polypropylene. Further proof of the presence of the side reaction is shown in the spectrums of unsulfonated and sulfonated high density polyethylene, given in Figure 5.8. In this figure, there is again strong absorbance at about 3200 cm'l, indicative of the sulfonic acid groups. The strong 89 :53' 44i- nI'I al.0- l I’ .—-.-l :44 ..i liié+d34Pédtii2 ll E-lil ‘i'ij- Spectrum of polypropylene film r 11.11; Me if I D Que .- 1 7.14:1, .l 70 ‘. l I one a" , V 0 In. JIJUUJ;GIJ.MI-:1.II... 12“ "I" . . .|.. .. . I . d‘ d00ll..l.lud..o.- . e . . a; .u. '1.‘ I'll!!! . . . m-T-I- ~I-u I «tum-4 -- 1101'“: b."- ImItl l I I. lbtdl Spectrum of sulfonated polypropylene film thelepeco'umsofunsmfonated andsutfonmdpolypropytene parison Figure 5.7 - Corn 90 1‘4.— 4:114”! 3m == EEbé-s- 113.311.1414 H-. II;- - - . . . . e-h 'mmv If" Mr .‘I‘ I “I 6. er .r'J-Jttr no 1!: Ian I l H” 7".1‘ "1:11.! 'ILILIL !.|': rrr- .-- - -~ W .rn Spectrum of HDPE film a...» m... .sin-.l.. l....An...-F u.- an.” ..._. .. . .- . . 1... . . an .- 414. . . 4.- .1111. n 4 r 1. le'lqli- .1...- 4....... .... - .. u”! H. .e . NO] "I; . ltrr. v1.1 ...m.4H|L.Hm.iw-.lml. lilim ii...- n...- . 1‘“ Spectrum of sulfonated HDPE film ’ and sulfonated HDPE Figure 5.8 - Comparison the IR spectrums of unsulfonated 91 absorbance at 1200 cm.‘1 is also due to the hydroxyl groups. But more important is the -C=C— stretching frequency at 1620 cm.’1 in the sulfonated HDPE spectrum. Once again it appears that the side reaction is an important factor in the sulfonation of this material. It is noted that the sulfonated sample in this case also had discolored after the reaction. On the other hand, Figure 5.9 shows the spectrums of unsulfonated and sulfonated polystyrene. These samples had been sulfonated for 20 minutes at the same concentration as polypropylene and HDPE. It was observed in the kinetic experiments that polystyrene was much more difficult to sulfonate than other materials and these spectra are indicative of that. There really is no difference between the spectra. Therefore, it is likely that if the reaction is to occur, it will be the addition of sulfonic acid groups. The probability of the side reaction occurring is quite small. It is therefore concluded that the hypothesis of the side reaction occurring for polypropylene and HDPE, but not for polystyrene, is valid. It also is likely that the side reaction will occur for other polymers in which the sulfonic acid group attacks the polymer backbone. 92 im im 4 _$ ii i#% gawfig“%q .. M3'1”:40111-114}...th win- Spectrum of polystyrene film 14444133114 Spectrum of sulfonated polystyrene film Figure 5.9 - Comparison the IR spectrums of unsulfonated and sulfonated polystyrene Chapter 6 Liquid Polymerization in a Thin Surface Film Analysis of an Epoxy-Diamine System: The system under consideration in this study is an epoxy coated fiber, as in a prepreg, which is embedded in an epoxy/diamine matrix. The matrix consists of a stoichiometric mixture of epoxy and diamine, and the fiber has an epoxy coating. The schematics of the system are shown in Figure 6.1. It is desired to know the characteristics of the diffusion of diamine into the epoxy coating and the subsequent reaction to cure the epoxy coating. In particular, it is hoped that a model of this system will shed light on the interphase region close to the fiber to determine if the epoxy is completely cured to the fiber interface or if a viscous liquid is present close to the fiber which allows movement of the fiber within the ‘matrix. This interphase region will have properties that are different from the cured epoxy in the matrix. To adequately model this system, both the diffusion of the diamine and the kinetics of the reaction must be well understood. With a good characterization of the diffusion and reaction in the interphase region, the properties of this region can be well understood. The epoxy used in this system is Epon 828 from Shell Chemical Company and the curing agent is metaphenylene 93 94 epoxy + amine epoxy < fiber coafing epoxy + anfine Figure 6.1 - Schematics of epoxy system 95 diamine. These species are shown in Figure 6.2. The reactants are mixed in stoichiometric portions and cured at 75' C for two hours, followed by curing at 125° C for two 29 hours. The reactions which occur during the cure of the epoxy are shown in Figure 6.3. It should be noted that both the epoxy and the diamine are tetrafunctional. In Figure 6.3, the reactions are divided between primary reactions and secondary reactions. In this study, only the primary reactions are considered because these reactions cause the crosslinking of the epoxy species and, hence, the cure. The secondary reactions do occur but are not as important in this study. In studies of this system conducted by Rao30, it was found that modification of the curing cycle was necessary since there was incomplete cure due to loss of the curing agent by diffusion at high temperatures. Rao found that a better curing cycle was: 24-36 hr 0 room temperature 2 hr 0 75' C 2 hr 0 125‘ C In this curing cycle, it has been suggested that the room temperature step allowed some of the reaction to occur and retarded the diffusion process. Rao found that the result was that the loss of curing agent at higher temperatures was reduced by this modification of the curing cycle. The objective of this study was to include reactant diffusion in a model for the curing process near the fiber 96 20mg 62.86 322856968 ~12 . 48 Sam A are 41.6 d \M/ _ \Ill us N .-IoIToI~:olo-1. Viol-A0 -o Ir~zo I_o|...:oln AU_ Lou-AW IU-Violfvlrolzo /0\ /.. \ (K . (ll. Hik 011—0 0 min. r Figure 6.2 - Reactants in epoxy system 97 Alla-.3..- 0-.-“: ..--. swam l\'CGLolIU|IO. R' 72 R' 4: I I I I NHn + (Tl—In —(‘,I-J. -——a> Ml-I—(TI-Iz—(tI-I—nb: \ / , 0 Primary Secondary Hydroxyl Amine epoxroe Amine R R H "it R fix I I I I . I . I—‘O—CH—CHa—NH+CH2—/CH ——-» Ho—CH—CHa—N—Cm—CH—OH _ \ - _ 0 . Secondary Te rtlary hydro x341 . E oxide . Amine p Amine Secondary Reactions: r- 5 r' R 9 : i I I I 1 NH *CH2—CH—OH 7" CH3 ‘_ CH *5" NH —CH p‘CH— W—CHTI— (if-i — (jig—g \ / ._ HYdIOXI/I .- . ' Et -er Hydroxyl epoxioe Figure 6.3 - Reactions occurring during the cure of the epoxy 98 and to determine if the room temperature step is as significant as Rao suggests. Rao found that the reaction was diffusion controlled at temperatures less than 80' C. Therefore, this work focused on the low temperature step of the curing process by providing a concentration-dependent diffusion coefficient for the diamine and rate constants for the reactions which occur at these low temperatures. Mechanisms for the Epoxy/Diamine Coupling: There has been quite a bit of discussion in the literature about the kinetics of the epoxy/diamine system, but there is little information about the diffusion of the diamine into the epoxy. Much of the kinetic information results from studies using differential scanning calorimetry (DSC). Use of data from the DSC assumes that the heat evolved is proportional to the extent of reaction. Kenny et. a1.31 compare their model of the processing of epoxy-based composites with DSC data. Their model describes the rheological behavior of commercial epoxy prepregs and they study the influence of different processing conditions. Tetra-Glycidyl Diamino Diphenyl Methane (TGDDM) and Diamino Diphenyl Sulfone (DDS) were the reactants in their system. They found that the temperature at the center of the laminate quickly reaches that of the external temperature,' due to the thermal conductivity of the fibers, and increases due to the imbalance between the rate of heat generation and 99 the thermal diffusivity of the composite. Other authors, such as Williams et. al.32, have found that for systems with lower conductivity there is less heat transferred to the fiber and the reaction is activated from the epoxy/epoxy-diamine interphase and moves to the epoxy/fiber interface. Neither of the authors discussed a diffusion coefficient for the curing agent but discussed a thermal diffusivity. Mijovic et. al.33 discuss a mechanistic modeling of epoxy-amine kinetics. They believe that the etherification reaction between the epoxy and hydroxyl groups often occurs at high temperatures, high degrees of cure, or in the presence of a large excess of epoxy groups. An important assumption in their work was that in the absence of Lewis acid or base type accelerators, and the temperature range used employed in the study (90°-120° C), the etherification and homopolymerization reactions are negligible. They also discuss various reaction routes for the formation of secondary and tertiary amines. By comparing the reaction mechanism for the formation of secondary and tertiary amines used in this thesis with those proposed by Mijovic et. al., it is found that the mechanisms that are used to model the process are the same as those predicted to occur over the majority of the cure cycle. They find that the epoxy groups initially interact with proton donor molecules to form a hydrogen bond complex. The reaction then occurs as: 100 (hydrogen bond complex) + amine ----> (transition complex) ----- > secondary or tertiary amine The transition complex is not explicitly included in the model used in this thesis, but implicitly the reaction to form the transition complex is included in the rate constant for the reaction. Buckley et. al.34 discuss DGEBA with 2,5-dimethyl 2,5-hexane diamine (DMHDA) as the curing agent and epoxy-cresol-novoloc resin. It is suggested that the primary amine hydrogens react with the epoxy at room temperature to form a linear polymer. The secondary amine does not react, however, until the sample is heated due to steric hindrance. These authors also indicate that the apparent activation energies for the curing reactions are similar, regardless of the initial condition. Their results are based upon Fourier transform infrared spectroscopy (FT-IR) and torsional braid analysis (TBA). Based on the above literature, several assumptions are included the model used in this thesis. The etherification reaction can be neglected for the temperature range being studied. Also, a complete cure of the system does not occur until the sample has been heated. This is indicated by the fact that the secondary amine does not react until the sample is heated. Thus, the reaction rate constants are temperature dependent. Also, the concentration of the hydrogen bond complex indicated by Mijovic et. al. is 101 assumed to be the same as the concentration of the epoxy coating the fiber. Equations Describing the System: The primary reactions which occur in this system were previously discussed. In short notation they are: A2 + E -—-> A3: k = k2 Eqn. 6.2 where A1 indicates the primary amine, A2 indicates the secondary amine, A3 indicates the tertiary amine, and E indicates the epoxy. With the above equations, the unsteady state material balances on this system are: 5.41 5 5111 =__ —- E ne 6e3 6t: 6x1)"1 6x kIAIE q .9_A_2=._5_D 5A2+k1AlE-k2A2E Eqn. 6.4 t 6x M 5x 6.43 5 6113 —=—D — E ne 6e5 I: 5x ‘1 5x +k2A2E q —6E--kAE-kAE E n 6 6 6t- 1 1 2 2 q . . The diffusion coefficients for the primary, secondary, and tertiary amine groups are concentration dependent and, therefore, included in the partial differential equation. Of great interest is the concentration of the tertiary amine, since the concentration of tertiary amine is directly 102 proportional to the degree of crosslinking in the system. But, of course, the tertiary amine cannot be formed until the primary and secondary amines are first formed. The diffusivity of the secondary and tertiary amines are slower than that of the primary amine, based upon molecular weight ratios. Since the epoxy is a large group, its diffusivity can be assumed to be zero and, therefore, the epoxy concentration changes with time due to the reactions only. Slab geometry is assumed in this model, since the problem is of diffusion in one direction only. Initially there is no diamine present in the system, so the initial conditions are: At t = 0; A1 = A10 = 0 A2==1Qm = 0 A3 = A30‘= 0 E = E0 Eqn. 6.7 where E0 is found from the molecular weight of the epoxy and the density. The boundary conditions are defined at x = 0, which is the epoxy/epoxy-diamine interphase and x = L, which is the epoxy/fiber interface. These boundary conditions are: At X = 0; A1 = All 103 —=0 Eqne 608 No further boundary conditions can be determined due to the reactions which are occurring. All can be determined from the molecular weight and the density of the diamine. Determination of the Solubility of m-xylene in Epoxy: Although there is a great deal of literature available which describes the kinetics of the epoxy/diamine system, there is very little information about the diffusion coefficient of the diamine into the epoxy. Therefore, to develop an accurate model for this system, the diffusion coefficient of the diamine was experimentally determined. Since the system is complicated because of the reaction which occurs, the diffusion coefficient was estimated by finding the diffusion coefficient of a similar material in the epoxy which does not react. The material used in this study to model the diffusion of the diamine was m-xylene. This probe molecule was assumed to have the same diffusion 104 volume as the diamine. The structure of this material is shown in Figure 6.4. The difference between the curing agent and m-xylene is that the amine groups are replaced by methyl groups. Therefore, no reaction will occur with the m-xylene, but it is approximately the same size and structure as the curing agent. As noted previously, the diffusivities of the secondary and tertiary amines (th and tbs) are lower that of the primary amine, due to the larger size of these molecules. DA2 and DA3 were estimated by multiplying the diffusivity of the primary amine by molecular weight ratios of the primary amine to the secondary or tertiary amine. Since diffusivity is known to be dependent upon molecular weight, this is a good assumption. To determine the diffusivity of m-xylene in the epoxy, an electrobalance was used, which measured the weight gain with time with a chart recorder. The epoxy was placed in a small aluminum pan so that only one side was exposed to the atmosphere. A schematic of the experimental setup is shown in Figure 6.5. Nitrogen was blown over the epoxy sample and the electrobalance measured the weight loss due to volatiles removed by the nitrogen. Once the epoxy sample reached a constant weight, it was ready for exposure to m-xylene. iM-xylene was converted to the vapor phase by bubbling nitrogen through the liquid m—xylene. The epoxy sample was then exposed to the vapor mixture of nitrogen and m-xylene 105 Figure 6.4 - Structure of m-xylene 106 \ Detector of 7 weight gain N2 + m-xylene atmosphere Aluminum pan with Epon 828 IV] V Figure 6.5 - Schematic of electrobalance experiment 107 and the weight gain with time was measured. The concentration of xylene at the gas-solid interphase was determined using the steady state weight gain in the epoxy sample. This experimentally determined concentration was checked by determination of the concentration through the use of solubility parameters. It is noted that these experiments were conducted at room temperature only and at this point a temperature dependent diffusion coefficient was not determined. The solubility of xylene in the epoxy was theoretically determined using the Flory-Huggins theory, which describes the entropy of mixing of polymer solutions. The entropy change of mixing of polymer molecules with small solvent molecules is: ASm=-R (xlln¢1+len¢2) Eqn . 6 . 9 Combining the entropy change on mixing with the non-ideal heat of mixing, the free energy change on mixing can be found. -AG R7? = (xllnct:1 +x21nd>2 ”(1902) Eqn. 6 ° 1° In equation 6.10, g is the polymer-solvent interaction jparameter. By differentiating equation 6.10 by n1, the number of moles of solvent, the reduced partial molar Gibbs 108 free energy of mixing for the solvent can be obtained, which then produces the solvent activity.37 A R? =1na1=1n¢1+¢2 ”(4,22 Eqn . 6 . 11 In equation 6.11, chi is the interaction parameter and the 6's represent the volume fractions. Once the solvent activity is determined, the solubility of xylene can be easily determined. To solve for the solvent activity, the interaction parameter must first be found. A useful equation to determine the interaction parameter is: V x=0'34+R_’:l‘(61-52)2 Eqn. 5°12 From the literature, it is known that the solubility parameter of m-xylene is 8.8 (cal/cm3)°'5 and its molar ‘volume is 0.1229 L/mol.28 The solubility parameter of Epon 828 epoxy is approximately 8.5 (cal/cm.3)°'5.38 Therefore, based upon equation 6.12, the interaction parameter is 0.3587. The volume fractions are determined from the molar volumes of each of the components and are; 61 = 0.2717 92 0.7283 Therefore, the solvent activity was determined to be 0.6808. 109 The solvent activity can be considered an effective mole fraction, such that; a1 = n(xylene)/n(epoxy) Since the mass of the epoxy was known to be approximately 0.085 g, the moles of epoxy were determined to be 2.219 x 10'42moles. Multiplying this number by the solvent activity gives 1.511 x 10'4 moles of xylene. The volume of the epoxy was found to be 7.398 x 10‘5 L, so the solubility of xylene in the epoxy was 2.0424 moles/L. This is the theoretical solubility of xylene in the epoxy. The electrobalance experiment was run until the weight gain reached steady state, which means that there was no more weight gain with exposure to the vapor. Figures 6.6a, b, and a show the results of three electrobalance experiments. Runs #1 and #2 were done at the same conditions, showing that the experiment was reproducible. Run #3 was done at a lower concentration of xylene vapor in the system. The concentration of xylene in the system could be checked with the theoretical value of 2.0424 mol/L by using the steady state weight gain found using the electrobalance. The concentration is the weight gain of xylene at steady state, divided by the molecular weight of xylene, divided by the volume of the epoxy sample. Since the volume of the epoxy sample was greater after exposure to xylene, due to 110 on. T 09 o: 8— db- )— III- 0505 08: on 00 om, 00 on b P p . q a q q 0 D 080.0 .80 0.8.0 ~80 008.0 08.0 0000.0 I\ --+- I---1——I—+—+— +--+-- f--+- - I—Ibu 6 (6) exoidn ssow 0.080 08.0 to It) 8. O + In 6 O “8.0 38.0 08.0 008.0 08.0 Figure 6.6a - Weight gain with time for run #1 using the electrobalance 111 G505 OE: 09 O. P 8_ 00 00 on 00 on O: 00 0m 0. 0 II 11 I 1: III I 1: II 11 IIIII it II... 1... II if III IIIIIIII 11 IIIIIIIII III-IIIIIIIIIIII- 4.. Figure 6.6b - Weight gain with time for run #2 using the electrobalance 220 240 260 280 200 180 140 160 time (hours) A I 120 80 1C0 II [Ill—Ll: L" ‘ I I éLII. 0.0106 - 0(1)]3 ‘r‘ 00312 - 00011 0031 OLIXJQ 0W8 0G1]? - A 6) axoidn ssow Figure 6.6c - Weight gain with time for run #3 using the electrobalance 113 the added xylene within the epoxy, the epoxy sample volume was 3 V (epoxy) = (mass epoxy)/(density epoxy) + (mass xylene)/(density xylene) The initial mass of the epoxy sample was 0.085 g, the density of the epoxy was 1.1625 g/cm3, and the density of m-xylene was 0.8642 g/cm3. The mass of xylene was taken as the maximum weight gain as indicated by the electrobalance. Experimentally determined concentrations of xylene are given in Table 6.1. I Run # Maximum weight Concentration of gain of xylene (g) xylene (mol/L) 1 0.008502 0.9653 2 0.008150 0.9298 3 0.001235 0.1593 Table 6.1: Experimental solubility of xylene n epoxy The concentrations reported in Table 6.1 are actually the experimental solubilities of xylene in the epoxy. These values are less than the theoretical solubility of 2.0424 mol/L. Theoretical solubilities typically have errors of 30-50% of the actual solubility, and therefore, are not very reliable. Thus, in modelling this system, the experimentally determined solubilities were used as the boundary condition at x = 0 to determine the diffusion coefficient of m-xylene in the epoxy, due to the greater confidence in these values. 114 Determination of the Diffusion Coefficient: The shape of these curves shown in Figures 6.6a, b, and c are typical for a system which is diffusion controlled. To determine the form of the diffusion coefficient of m- xylene into the epoxy, the following partial differential equation was solved numerically: 60' 5 6c; —£=._— _ E e 6.13 6t 6xDCx 6x qn where Cx is the concentration of xylene. The method used to solve this equation is the same as that used to solve Equation 6.3, except that no reaction occurs in this process. The output of the model was concentration of xylene along length increments of the cross section of the film. To determine the mass uptake from the model results, the concentrations of xylene across the depth of the sample were averaged and multiplied by the volume of the epoxy sample and the molecular weight of m-xylene. The volume of the epoxy sample was the sum of the volume of epoxy and the :m-xylene which has diffused into the epoxy. Initially, a constant diffusion coefficient was used to fit the experimental data. Although a constant diffusion «coefficient would fit the experimental curves at low times, 'the difference was too great at long times. Also, the constant diffusion coefficient which fit the data at low times for run #3 was 2 x 10'12 mz/sec, while the diffusivity 115 for run #2 was 6.3 x 10’12 m2/sec. Comparisons of the constant diffusivity to the experimental results are shown in Figure 6.7a and b, which show how the constant diffusion coefficients fit the data well until high times. Since the diffusivities were different for runs with different concentrations, however, it was concluded that diffusion coefficient was dependent upon concentration. ‘A common form of the concentration-dependent diffusivity has an exponential dependency upon concentration. This form of this diffusivity is: D=Doe°c Eqn. 6 . 14 where Do is the diffusion coefficient in the limit of zero concentration and a is an arbitrary constant. Since the constant diffusion coefficients which fit the data at low concentrations were known, it was necessary to determine the constants of the above equation which gave approximately the same diffusion coefficients at low xylene concentrations. After several attempts to fit the data, it was found that the best fit for both runs were found with; D0 = 1.5 x 10'12 mz/sec i- 0.4 x 10'12 a = 1.69 i 0.2 A comparison of the exponential diffusion coefficient with the constant diffusivity and the experimental results are shown in Figure 6.7a and b. Theses figures shows that the 603 0E: gov 808w 800w gm gnu gm 880. g. 800 0 _ _ s r b b > f h p q A a. . d u 4 a u 116 QoovoxoVOOuo . N _ .wm.ouo o _oEmEcmaxw Il-l $3. 0(5 38.0 8.6 i 28.0 3.3-3.3.3.... 1m $8.0 $8.6 ) exoidn ssow Figure 6.7a - Comparison of experimental results to model results for run #1 117 20898090 m—bmuo IIOII _oEoEzocxw ll-Il “00$ 05: 0 SHE—gogogmggmgvggg— — h b — q a p — n 0 m. In... 0' (6) exoidn ssou: g .80 _ .80 98.0 98.0 Figure 6.7b - Comparison of experimental results to model results for run #3 118 exponential dependency on concentration provides a good fit to the experimental data, except at very high times. Since the fit to the experimental data is good over a majority of the time range, it was taken as a good approximation to the concentration-dependent diffusion coefficient, particularly for use in the mathematical model of this thesis. A least squares error analysis was done on both the constant diffusion coefficient results and the exponential dependency results. For run #2, the least squares error for the constant diffusivity was 1.33 x 10‘6, while the least squares error for the exponentially concentration dependent diffusivity was 1.9 x 10'7, almost an order of magnitude less, indicating a better fit to the experimental data. For run #3, the least squares error for the constant diffusivity was 7.44 x 10'8, while the error for the exponentially concentration dependent diffusivity was 1.12 x 10'7. Table 6.2 summarizes the results of the least squares analysis. Although the error for the exponential form of the diffusivity is slightly higher for run #3, it is still quite low and the form of the diffusion coefficients, with the values of Do and a given above, provide a good approximation to the diffusion coefficient of the curing agent into the epoxy at low temperatures. 119 Run # Least Squares Least Squares Error for Constant Error for D Exponential D 2 1.33 x 10’6 1.90 x 10'7 3 7.44 x 10'8 1.12 x 10'7 T C 6.2 " Compar 8011 0 L638 Squares Error i0! Constant and Exponentially-Dependent Diffusivities In the above determination of the diffusion coefficient, the temperature dependence of the diffusion coefficient was not addressed. In reality, the diffusion coefficient has the form; .E D=D0Aexp (aC) exp(--F;-,) Eqn. 5- 15 Since the experiments described above were run at room temperature (25' C), DoA can be determined by solving the following equation; 2 E Eqn. 6.16 D =1.5x10‘12-5n—-= ex -—" ° sec °“ p( RT) The resulting value for D0A is 3.24 x 10'5. Determination of reaction rate constants: The next step in the modelling of the diffusion and reaction problem was the determination of the kinetic rate constants for the two primary reactions which occur. Rate constants have been reported for high temperature cures, 120 where the reaction is not diffusion-controlledso, but there is not much information about rate constants for low temperature cures (less than 80' C). However, Finzel35 reported that the rate constant for the second reaction was one-half that of the rate constant for the first reaction. Therefore, the rate constants can be determined by fitting the model to experimental data. It is assumed in the model that the rate constants have an Arrhenius dependence upon temperature, or: -Ed ——' Eqn. 6.17 .. T k-koexp R where Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. To solve the problem, k was'varied until a good fit with the experimental data was obtained. It was assumed that E3 was a constant value. Before the model can be used to find the rate constants, the initial and boundary conditions of the system ‘were specified. The molecular weight of the epoxy resin was 383 g/gmol and the density of the resin was 1.1625 g/cm3. The molecular weight of mPDA was 108 g/gmol and its density ‘was 1.0686 g/cm3. At t = 0, only the epoxy surrounded the fiber, so the concentration of epoxy groups was; 121 93 EO=1000 2 =1.5176$’L£1— Eqn. 6.18 where the 2 takes into account the fact that there are two epoxide groups within each resin chain and the 1000 is a conversion factor so that the concentration is in units of gmol/L. In the matrix surrounding the epoxy layer there was a 1:1 stoichiometric ratio of amine to epoxy and 14.5 parts by weight amine to 100 parts by weight of epoxy resin at t>0. The volume of this matrix was; 14.5+1oo A 93 v=1000( )=0.0996L Eqn. 6.19 Therefore, the concentration of primary amine groups at x = 0 was; 14.5 1 Ego} A = =103464 . . 11 ( MWA ) 0.0996L L Eqn 6 20 As noted earlier in this chapter, the curing cycle was modified to allow reaction at room temperature to slow the diffusion of diamine through the epoxy. To account for this in the model, both Eo and All are multiplied by l-X, where X is the extent of reaction during the room temperature portion of the cure. If this is not accounted for in the model, diffusion will be too fast and the mathematical model 122 will overflow. This would be an indication that the equations describing the system above are invalid and the process is not diffusion-controlled. However, the system has been shown to be diffusion-controlled below 80° C, so inclusion of the extent of reaction during room temperature cure is important. In this model, X was estimated to be 0.30. The experimental data that the model attempts to fit was determined by Rao30 and is the extent of reaction versus time. A plot of this data is given in Figure 6.8. Since two reactions occur in the process, the best method of determining the extent of cure is by monitoring the epoxide concentration. Therefore, the extent of reaction is; CEO ' C8 C80 X= Eqn. 6.21 where C86 is the concentration of epoxide groups at t = 0 and CE is the concentration of epoxide groups at x = 0 and time t. The gel point can be determined by; _ 1 Pc‘ .1. Eqn. 6. 19 (1+p(f-2))2 where f is the functionality of the branch unit and rho is the ratio of all A groups, reacted and unreacted, that are Part of the branch units, to the total number of A groups in 123 .56.. GE: 00m 00p omp :5 0m. 0 0p 00 0: CV ON C . l .lfi.-- -. i ll il.l.i_i|lii m-lliil_i ixluvmn \D..\ .\ .—.. \C\ \- c\ \\ I .r _ -..-l- _lll..i|*i.|. ii._-l ii. i-.- .llilill il E \ .7 6 ca 266.6 I -6- i \ a \ \1 .hu n _ 08:33 i -.-l t \ _ii: i .i -i -l i. .2! 2:. \ -\Cl i_.\ \ - .. mvd 0 mod P0 who N0 mud 0". 0 mod v 0 names: go suezxe moo 1.. "C Figure 6.8 - Experimental extent of reaction 124 41 the mixture. In this case the diamine is the branch unit and is tetrafunctional. Therefore, f = 4. If A is the epoxide group, then there are 2 epoxide groups involved in the branch and 4 which are in the mixture. Therefore, rho is 0.5. Thus, the critical extent of reaction, which occurs at the gel point, is 0.7071. The model is able to predict the gelation time using this critical extent of reaction. From the data by Rao, a critical extent of reaction of 0.7071 is not obtained during the low temperature portion of the cure cycle. Temperatures higher than 80’ C are required to reach gelation. To obtain the curves for the calculated extents of reaction shown in Figure 6.8, the reaction rate constants were estimated to obtain an approximate fit to the experimental data. At 80° C the best fit rate constant was 0.000178 L/mol, while at 50° C the rate constant was 0.000076 L/mol. A fit of the model results to the experimental extents of reaction is shown in Figure 6.9. To determine the temperature dependency of the rate constant, ln ko versus l/T was plotted, as in Figure 6.10. The slope of the plot was -2347.18 = —Ea/R. Therefore, the activation energy for the reaction was 4.664 kcal/mol. The y-intercept of the line was -2.1325 = ln k0. Therefore, k.o was found to be 0.1185 1' 0.005. Finally, the concentration gradients for the primary iamine, secondary amine, and tertiary amine within the epoxy 125 l-i'| 6 as 62.8 .86: 6 as .62.... 6 8. 32.8 6 08 E063 3002 .. 00w Tit--. T e- T- 00¢ 02 03 9:555:53. on -_. on... 8' il'l—i.|il I..# II 1. .- . no.0 . 1' 0° no.0 Figure 6.9 - Comparison of model to experimental extent of reaction 126 000.0 0N000 ii .I..T--- N000 .. 0.0. v o. t 0.0. t 0.5. 1. h. . 0.0. ./. . . ”.0. ..n? ,/ . 0.0. /.. 5 0. /... 4 0d. 0.? PI 050.0 50.0 0000.0 0.0. (WU Ut Figure 6.10 - Plot to determine kO and Ba 127 layer after two hours of exposure at 75° C are shown in Figures 6.11, 6.12, and 6.13. The concentration of the primary amine decreases with position into the epoxy as expected. However, the concentration of both the secondary and tertiary amine species increases with position into the epoxy. A possible explanation for this phenomenon may be that the diffusion of the primary amine is faster than the reaction mechanism, so the diffusing species does not react until it reaches layers farther from the interface. This correlates well with the earlier statement that this is a diffusion controlled process at temperature less than 80° C. Further studies To provide a more accurate determination of the diffusion coefficient and the kinetic rate constants at temperatures less than 80° C, further experimental data is required. In this research, experimental data was only available at 50° C, 80° C, and higher temperatures. To further determine the characteristics of the diffusion controlled region, two or more sets of data less than 80° C are required. The model used in this thesis did not consider the import of the secondary reactions, and further studies of 'these reaction mechanisms are required. 128 I I I . I ' | . i ' 120 140 160 130 200 I 100 Position (microns) 0 942 O 941 0.94 0.939 0 938 0 937 0.936 0 931 0 93 0 929 0.928 0 927 (mom) uoguuuesuoa Figure 6.11 - Predicted concentration primary amine after two gradient of hours at 75' C 129 . I I i 9 i 160 180 200 I i 140 I 100 Position (microns) 0 075 - 0 07 0 065 0 06 0 055 0 05 0.045 0 04 0 035 0‘ 0 025 0 02 0.015 0 01 0.005 (mow) uouuwnuoa Figure 6.12 - Predicted concentration of secondary amine after two hours at 75' C 130 . I I I I I i ) o g a 100 120 140 160 180 200 Position (microns) 40 80 I i I o 000 , 0 0055 0 005 l 0 0045 . 0 0 0035 0 003 0 0025 3 0 ooms 00m 0 0005 (mow) uottutuaouog Figure 6.13 - Predicted concentration gradient of tertiary amine after two hours at 75' C APPENDICES 132 APPENDIX A - Basic Program for Modelling Diffusion The following basic program was written to model the diffusion of a vapor species into a solid, polymeric material. It is assumed that the diffusion coefficient is constant and the solid-vapor boundary does not swell with time. To model this system a explicit finite difference method was used, the Saul’yev method. The differential equations describing the system can be solved using LaPlace transforms and the result is an error function equation. The output from this model was compared to the error function to determine the accuracy of the model. ’Test of model to compare to error function OPEN "GMM.DAT" FOR OUTPUT AS #1 'QUESTIONS OF THE USER PRINT "WHAT IS THE POSITION INCREMENT IN METERS?": INPUT DELX# PRINT "WHAT IS THE DIFFUSIVITY OF THE GAS IN M‘Z/SEC?": INPUT D# PRINT "WHAT IS THE TIME STEP IN SECONDS?": INPUT DELT# PRINT "WHAT IS THE MAXIMUM PENETRATION DEPTH IN MICRONS?”: INPUT INDEX PRINT "AT WHAT POSITIONS WOULD YOU LIKE A PRINTOUT?": INPUT A: INPUT B: INPUT C: INPUT D: INPUT E ’SET ARRAY SIZE MARK = 10 REPS = 10 LET SIZE = INDEX + 1 DIM V#(INDEX, INDEX) DIM P#(INDEX, INDEX) DIM Q#(INDEX, INDEX) COUNT = 0 'INITIAL CONDITIONS FOR I = 0 TO INDEX 133 P#(I. 0) Q#(I. 0) V#(I. 0) NEXT I "II" 000 ’BOUNDARY CONDITIONS FOR N = 1 To MARK P#(0: N) Q#(0. N) Q#(INDEX, N V#(0: N) NEXT N 3 3 ) 3 LET B# = (D# * DELT#) / ((DELX#) A 2) FOR J 1 TO REPS FOR N 0 TO MARK - 1 ’CALCULATION FOR GAS CONCENTRATION FOR I = 1 TO INDEX - 1 P#(I, N + 1) = (B# * (P#(I - 1, N + 1) - V#(I, N) + V#(I + 1, N)) + V#(I, N)) / (1 + B#) NEXT I FOR I = 1 TO INDEX - 1 Q#(INDEX - I, N + 1) = (B# * (V#(INDEX - I - 1, N) - V#(INDEX - I, N) + Q#(INDEX - I +1, N + 1)) + V#(INDEX - I, N)) / (1 + B#) NEXT I FOR I = 1 To INDEX V#(I, N + 1) = .5 * (P#(I, N + 1) + Q#(I, N + 1)) NEXT I NEXT N ’OUTPUT INSTRUCTIONS 900 COUNT = COUNT + 1 PRINT A * DELX#; "METER", B * DELX#; "METER", C * DELX#; "METER", D * DELX#; "METER", E * DELX#; "METER" FOR N = 1 TO MARK PRINT V#(A, N); V#(B, N); v#(c, N); V#(D, N); V#(E, N) WRITE #1. V#(A. N). V#(B. N). V#(C, N). V#(D: N),V#(E:N) NEXT N GOSUB 900 PRINT "COUNT= "; COUNT * DELT# * 10; "SEC" NEXT J CLOSE #1 END FOR I = 0 TO INDEX V#(I, 0) = V#(I, MARK) p#(I, 0) = P#(I, MARK) Q#(I. 0) = Q#(I: MARK) NEXT I RETURN 134 APPENDIX B - Basic Program to Model Sulfonation The following Basic computer program is a modification Of the program in appendix A, which includes the concentration-dependent diffusion coefficient for sulfur trioxide vapor and the kinetic rate constant. In the program, V# denotes the concentration of sulfur trioxide vapor, z# is the concentration of unsulfonated polymer, S# is the concentration of the sulfonated polymer, and MT# is the mass uptake of sulfur trioxide vapor in the polymer. 'Diffusion of sulfur trioxide gas into polymer samples OPEN ”8032.CSV" FOR OUTPUT AS #1 ’QUESTIONS OF THE USER PRINT "WHAT IS THE POSITION INCREMENT IN METERS?": INPUT DELX# PRINT "WHAT Is THE TIME STEP IN SECONDS?": INPUT DELT# PRINT "WHAT IS THE MAXIMUM PENETRATION DEPTH IN MICRONS?": INPUT INDEX PRINT "WHAT IS THE TEMPERATURE OF THE SYSTEM? (IN DEGREES CELSIUS)": INPUT T# PRINT "WHAT IS THE WEIGHT OF THE POLYMER SAMPLE? (IN GRAMS)": INPUT WT# PRINT "WHAT IS THE MOLECULAR WEIGHT OF THE POLYMER FILM?": INPUT MW# PRINT "WHAT IS THE VOLUME OF THE POLYMER SAMPLE? (IN LITERS)": INPUT VOL# PRINT "AT WHAT POSITIONS WOULD YOU LIKE A PRINTOUT?": INPUT A: INPUT B ’SET ARRAY SIZE TP# = T# + 273.15 MARK = 10 REPS = 60 DIM V#(INDEX, INDEX) DIM P#(INDEX, INDEX) DIM Q#(INDEX, INDEX) DIM Z#(INDEX, INDEX) DIM Zl#(INDEX, INDEX) DIM S#(INDEX, INDEX) DIM Sl#(INDEX, INDEX) 135 DIM MT1#(INDEX, INDEX) DIM MT2#(INDEX, INDEX) DIM MT#(INDEX, INDEX) COUNT = 0 LET To = 0 LET M = 0 ’INITIAL CONDITIONS FOR I = 0 TO INDEX Z#(I, 0) = DEN# * 1000 / MW# DEN# * 1000 / MW# Z1#(I, 0) = S#(I, 0) = 0 Sl#(I, 0) = 0 P#(I. 0) Q#(I: 0) V#(I: 0) NEXT I O 0 0 'BOUNDARY CONDITIONS FOR N = 1 To MARK Z#(INDEX, N) = S#(INDEX, N) = 0 P#(O, N) = 2.5022 P#(INDEX, N) = 0 Q#(O, N) = 2.6022 Q#(INDEX, N) = 0 v#(0, N) = 2.6022 V#(INDEX, N) = 0 NEXT N DO# A= K0# EA# R: K# B# FOR J = FOR N = = 1.37E-14 .41 = 2.22 = 10000 8.314 * .239 1 TO REPS 0 TO MARK - 1 FOR I = 1 TO INDEX - 1 DEN# * 1000 / MW# K0# * EXP(-EA# / (R * TP#)) (DO# * DELT#) / ((DELX#) A2) 'CALCULATION FOR GAS CONCENTRATION P#(I, N + 1) = (V#(I, N) + B# * EXP(A * V#(I, N)) * (P#(I - 1, N + 1) - V#(I, N) + V#(I + 1, N)) + B# * A * EXP(A * v#(I, N)) * (V#(I: N) * V#(I: N) ' 2 * V#(I + 1, N) * V#(I, N) + V#(I + 1, N) * V#(I + 1, N)) - K# * DELT# * v#(I, EXP(A * v#(I, N))) NEXT I N) * Z#(I, N)) / (1 + B# * P#(INDEX, N + 1) = (V#(INDEX, N) + B# * EXP(A * V#(INDEX, N)) * (P#(INDEX - 1, N + 1) - V#(INDEX, N) + v#(INDEX - 1, N)) + B# * A * EXP(A * V#(INDEX, 136 N)) * (V#(INDEX, N) * V#(INDEX, N) — 2 * v#(INDEX - 1, N) * V#(INDEX, N) + V#(INDEX - 1, N) * v#(INDEX - 1, N)) - K# * DELT# * V#(INDEX, N) * Z#(INDEX, N)) / (1 + B# * EXP(A * v#(INDEX, N))) Q#(INDEX, N + 1) = (V#(INDEX, N) + B# * EXP(A * V#(INDEX, N)) * (V#(INDEX - 1, N) - V#(INDEX, N) + Q#(INDEX - 1, N)) + B# * A * EXP(A * V#(INDEX, N)) * (V#(INDEX, N) * V#(INDEX, N) - 2 * V#(INDEX - 1, N) * V#(INDEX, N) + V#(INDEX - 1, N) * V#(INDEX - 1, N)) - K# * DELT# * V#(INDEX, N) * Z#(INDEX, N)) / (1 + B# * EXP(A * V#(INDEX, N))) FOR I = 1 TO INDEX - 1 Q#(INDEX - I, N + 1) = (V#(INDEX - I, N) + B# * EXP(A * V#(INDEX - I, N)) * (V#(INDEX - I - 1, N) - V#(INDEX - I, N) + Q#(INDEX - I + 1, N + 1)) + B# * A * EXP(A * V#(INDEX - I, N)) * (V#(INDEX - I, N) * V#(INDEX - I, N) - 2 * V#(INDEX - I + 1, N) * V#(INDEX - I, N) + V#(INDEX - I + 1, N) * V#(INDEX - I + 1, N)) - K# * DELT# * v#(INDEX - I, N) * Z#(INDEX - I, N)) / (1 + B# * EXP(A * V#(INDEX - I, N))) NEXT I M = M + 1 T = To + M * DELT# FOR I = 1 TO INDEX V#(I, N + 1) = .5 * (P#(I, N + 1) + Q#(I, N + 1)) NEXT I ’CALCULATION OF UNSULFONATED POLYMER CONCENTRATION FOR I = 1 TO INDEX Z#(I, N + 1) = Zl#(I, 0) - K# * DELT# * Z#(I, N) * V#(I, N) NEXT I ’CALCULATION OF SULFONATED POLYMER CONCENTRATION FOR I = 1 TO INDEX S#(I, N + 1) = S1#(I, 0) + K# * DELT# * Z#(I, N) * V#(I, N) NEXT I MT1#(0, N + 1)= (2 * (.5 * V#(0, N + 1) + V#(1, N + 1) + V#(2, N + 1) + V#(3, N + 1) + V#(4, N + 1) + V#(5, N + 1) + V# #(6, N + 1) + v#(7, N + 1) + V#(8, N + 1) + v#(9, N + 1) + V#(10, N + 1) + V#(11, N + 1) + v#(12, N +1) + v#(13 , N + 1) + v#(14, N + 1) + v#(15, N + 1) + V#(16, N + 1) + v#(17, N + 1) + V#(18, N + 1) + V#(19, N + 1) + V#(20, N + 1) + V#(21, N + 1) + V#(22, N + 1) + V#(23, N + 1) + v#(24, N + 1) + V#(25, N + 1) + V#(26, N + 1) + v#(27, N + 1) + V#(28, N + 1) + v#(29, N + 1) + * V#(29, N + 1)) / 31) * (VOL# + MT#(0, N) / (1000 * 1.97)) * 80.06 137 MT2#(0, N + 1) = (2 * (.5 * S#(o, N + 1) + S#(1, N + 1) + S#(2, N + 1) + S#(3, N + 1) + S#(4, N + 1) + S#(5, N + 1) + S#(6, N + 1) + S#(7, N + 1) + S#(8, N + 1) + S#(9, N + 1) + S#(10, N + 1) + s#(11, N + 1) + S#(12, N + 1) + S#(13, N + 1) + S#(14, N + 1) + S#(15, N + 1) + s#(16, N + 1) + S#(17, N + 1) + S#(18, N + 1) + s#(19, N + 1) + S#(20, N + 1) + s#(21, N + 1) + s#(22, N + 1) + S#(23, N + 1) + S#(24, N + 1) + S#(25, N + 1) + S#(26, N + 1) + S#(27, N + 1) + S#(28, N + 1) + S#(29, N + 1) + .5 * S#(29, N + 1)) / 31) * (VOL# + MT#(0, N) / (1000 * 1.97)) * 80.06 HT#(O, N + 1) = MT1#(O, N + 1) + MT2#(O, N + 1) NEXT N 'OUTPUT INSTRUCTIONS 900 COUNT = COUNT + 1 PRINT A * DELX#; "METER", B * DELX#; "METER" FOR N = 1 TO MARK PRINT V#(B. N). Z#(B. N). S#(B. N). MT#(0. N) WRITE #1, MT#(0, N) NEXT N GOSUB 900 PRINT "COUNT= "; COUNT * DELT# * 10; "SEC" NEXT J CLOSE #1 END FOR I = 0 TO INDEX V#(I, 0) = V#(I, MARK) P#(I. 0) = P#(I. MARK) Q#(I. 0) = Q#(I, MARK) Z#(I, 0) = Z#(I, MARX) s#(I, 0) = s#(I, MARK) MT#(I. 0) = MT#(I. MARK) NEXT I RETURN 138 APPENDIX C - Weight: Gain from Sulfonation A l B 1 c I D I E LF I G I H _1_ Weight gainlin polystyrene and lypropylene due to sulfonation with sulfur triloxide 2 3 Run #1 - PS Run #1 - PP Run #2 - PS 4 5 Reactor T 90 Reactor T 90 Reactor T 90 6 Bath T 11 Bath T 11 Bath T 20 7 Wt 0.0463 L Wt= 0.0501 g Wt 0.0464 g 8 9 10 Time (sec) Wt gain (9) Time (sec) Wt gain (9) Time (sec) Wt gain 11 0 0 0 0 0 0 12 30 0.0004 30 30 0 13 60 0.00073 60 0.0003333 60 0 14 90 0.000867 90 0.0028 90 0.000867 15 120 0.000933 120 0.0028667 120 0.001 16 150 0.000933 150 0.0048667 150 0.001267 17 180 0.001067 180 0.005 180 0.001 18 210 0.001067 210 0.0074667 210 0.001133 19 240 0.001267 240 0.0092 240 0.001133 20 270 0.000933 270 0.0106667 270 0.001133 21 300 0.001067 300 0.0118667 300 0.001133 22 330 0.001 330 0.0128667 360 0.0014 23 360 0.001133 360 0.0137333 420 0.00133 24 390 0.001267 390 0.0152 450 0.001533 25 420 0.001333 420 0.0161333 26 450 0.001267 450 0.0168667 27 480 0.001133 480 0.0175333 28 510 0.001333 29 540 0.001067 30 570 0.001067 31 630 0.001467 32 690 0.001067 33 34 35 36 37 38 39 40 41 139 I J L M O P 1 2 3 Run#2-PP Run#3-PS Run#3-PP 4 5 ReactorT 90 ReactorT 110 ReactorT 110 6 Bath T 20 Bath T 11 Bath T 11 7 Wt= 0.04999 Wt= 0.0456 g Wt= 0.051 g 8 9 10 Time (sec)tha_in Lg) Time (sec)thain (9) Time (sec) Main (g)_ 11 0 0 0 0 0 0 12 30 30 30 0.0008 13 60 60 60 0.0055333 14 90 90 90 0.0091333 15 120 0.001 120 0.0002 120 0.0126 16 150 0.0023333 150 0.0002667 150 0.0158 17 180 0.0036667 180 0.0002667 180 0.0188667 18 210 0.0047333 210 210 0.0218 19 240 0.0060667 240 0.0002667 240 0.0238 20 270 0.0076 270 0.0002667 270 0.0268 21 300 0.0088667 300 0.0002 300 0.0282 22 330 0.0106 330 0.0002 330 0.0307333 23 360 0.0115333 360 0.0004 360 0.0329333 24 390 0.0128 390 0.0004 390 0.0347333 25 420 0.0142667 420 0.0005333 420 0.0370667 26 450 450 0.0005333 450 0.0384667 27 480 0.0167333 480 0.0006 480 0.0404667 28 510 0.018 510 0.0006 510 0.042 29 540 0.0192 540 0.0006 540 0.0446 30 570 0.0203333 570 0.0008 570 0.0471333 31 600 0.0215333 600 0.0007333 600 0.0485333 32 630 0.0228 660 0.0007333 33 660 0.024 720 0.0011333 34 690 0.0256 780 0.0012 35 720 0.0266 840 0.0013333 36 900 0.0014667 37 960 0.0015333 38 1020 0.0016 39 1080 0.0015333 40 1140 0.0017333 41 1200 0.0018667 140 Q R T U 1 2 3 Run#4-PS Run#4-PP 4 5 ReactorT 110 ReactorT 110 6 Bath T 20 Bath T 20 7 Wt= 0.0445; Wt= 0.0499 8 9 10 Time (sec) Wt Jgain (9) Time (sec) WLgang) 11 0 0 0 0 12 30 0.0004 30 13 60 0.000867 60 0.0028 14 90 0.000933 90 0.006533 15 120 0.001 120 0.0112 16 150 0.001067 150 0.014667 17 180 0.0012 180 0.017067 18 210 0.0012 210 0.0204 19 240 0.001333 240 0.023 20 270 0.001533 270 0.025667 21 300 0.0014 300 0.0284 22 330 0.001533 330 0.030667 23 360 0.001733 360 0.033133 24 390 0.001933 390 0.0354 25 420 0.001933 420 0.037667 26 450 0.001867 450 0.039733 27 480 0.002 480 0.0418 28 510 0.0018 510 0.043333 29 540 0.0018 540 0.045133 30 570 0.001867 570 0.046867 31 600 0.001933 600 0.048667 32 660 0.001933 33 720 0.001933 34 780 0.002 35 840 0.0022 36 900 0.002267 37 960 0.002333 38 1020 0.0024 39 1080 0.002467 40 1140 0.0026 41 1200 0.002867 141 APPENDIX.D - Basic Progran.to Model Curing of an Epoxy This program is also a modification Of the program in Appendix A to model the diffusion and reaction of a diamine into an epoxy system. In this program, V# is the concentration Of the primary amine, E# is the concentration Of the uncured epoxy, DM# is the concentration of the secondary amine, and MDM# is the concentration of the tertiary amine. 'Model for the curing Of an epoxy resin system OPEN ”AMI.CSV" FOR OUTPUT AS #1 'QUESTIONS OF THE USER PRINT "WHAT IS THE POSITION INCREMENT IN METERS?": INPUT DELX# PRINT "WHAT IS THE TIME STEP IN SECONDS?": INPUT DELT# PRINT "WHAT IS THE MAXIMUM PENETRATION DEPTH IN MICRONS?": INPUT INDEX PRINT "WHAT IS THE TEMPERATURE OF THE SYSTEM? (IN DEGREES CELSIUS)": INPUT T# PRINT "AT WHAT POSITIONS WOULD YOU LIKE A PRINTOUT?": INPUT A: INPUT B 'SET ARRAY SIZE DELX# - 6.667E-06 DELT# - 10 INDEX - 30 T# - 80 TP# - T# + 273.15 MARX - 10 REPS - 76 DIM V#(INDEX, INDEX) DIM P#(INDEX, INDEX) DIM Q#(INDEX, INDEX) DIM MT#(INDEX, INDEX) DIM Z#(INDEX, INDEX) DIM Zl#(INDEX,INDEX) DIM Zl#(INDEX, INDEX) DIM DM#(INDEX, INDEX) DIM DM1#(INDEX, INDEX) DIM DM2#(INDEX, INDEX) DIM MTM#(INDEX, INDEX) DIM MTM1#(INDEX, INDEX) DIM MTM2#(INDEX, INDEX) DIM X#(INDEX, INDEX) DIM X1#(INDEX, INDEX) COUNT - o 142 LET T0 - 0 LET M - 0 'INITIAL CONDITIONS FOR I - 0 TO INDEX V#(I, 0) - 0 P#(I, 0) - 0 Q#(I. 0) - 0 MT#(O, 0) - 0 Z#(I, 0) - 0.7 * 1.5176 Z1#(I, 0) - 0.7 * 1.5176 DM#(I, 0) - 0 DM1#(I, 0) - 0 DM2#(I, 0) - 0 DM#(I, 0) - 0 DM1#(I, 0) - 0 DM2#(I, 0) - o X#(O, 0) - 0 NEXT I 'BOUNDARY CONDITIONS FOR N - 1 TO MARK V#(O, N) - 0.7 * 1.3464 P#(0, N) - 0.7 * 1.3464 Q#(0, N) - 0.7 * 1.3464 NEXT N LET ED# - 10000 LET R# - 8.314 * 0.239 00# - 1.5E-12 DOB# - 0.0000324 DOA# - DOB# * EXP(-ED# / (R# * T1#)) LET B# - (DOA# * DELT#) / (DELX#) “2 A - 1.69 Kl# - 0.000178 K2# - 0.5 * K1# FOR J - 1 TO REPS FOR N - 0 TO MARK - 1 'CALCULATION FOR PRIMARY AMINE CONCENTRATION FOR I - 1 TO INDEX - 1 P#(I, N + 1) - (V#(I, N) + B# * EXP(A * V#(I, N)) * (P#(I - 1, N + 1) - V#(I, N) + V#(I + 1, N)) + B# * A * EXP(A * V#(I, N)) * (V#(I, N) * V#(I, N) - 2 * V#(I + 1, N) * V#(I, N) + V#(I + 1, N) * V#(I + 1, N)) - K1# * DELT# * V#(I, N) * Z#(I, N)) / (1 + B# * EXP(A * V#(I. N))) NEXT I P#(INDEX, N + 1) - (V#(INDEX, N) + B# * EXP(A * V#(INDEX, N)) * (P#(INDEX - 1, N + 1) - V#(INDEX, N) + V#(INDEX - 1, N)) + B# * A * EXP(A * V#(INDEX, N)) * (V#(INDEX, N) * V#(INDEX, N) - 2 * V#(INDEX - 1, N) 143 * V#(INDEX, N) + V#(INDEX - 1, N) * V#(INDEX - 1, N)) - K1# * DELT# * V#(INDEX, N) * Z#(INDEX, N)) / (1 + B# * EXP(A * V#(INDEX, N))) Q#(INDEX, N + 1) - (V#(INDEX, N) + B# * EXP(A * V#(INDEX, N)) * (V#(INDEX - l, N) - V#(INDEX, N) + Q#(INDEX - 1, N)) + B# * A * EXP(A * V#(INDEX, N)) * (V#(INDEX, N) * V#(INDEX, N) - 2 * V#(INDEX - 1, N) * V#(INDEX, N) + V#(INDEX - 1, N) * V#(INDEX - 1, N)) - Kl# * DELT# * V#(INDEX, N) * Z#(INDEX, N)) / (1 + B# * EXP(A * V#(INDEX, N))) FOR I - 1 TO INDEX - 1 Q#(INDEX - I, N + 1) - (V#(INDEX - I, N) + B# * EXP(A * V#(INDEX - I, N)) * (V#(INDEX - I - 1, N) — V#(INDEX - I, N) + Q#(INDEX - I + l, N + 1)) + B# * A * EXP(A * V#(INDEX - I, N)) * (V#(INDEX - I, N) * V#(INDEX - I, N) - 2 * V#(INDEX - I + 1, N) * V#(INDEX - I, N) + V#(INDEX - I.+ 1, N) * V#(INDEX - I + 1, N)) - K1# * DELT# * V#(INDEX - I, N) * Z#(INDEX - I, N)) / (1 + B# * EXP(A * V#(INDEX - I, N))) NEXT I M - M + 1 T - TO + M * DELT# FOR I - 1 TO INDEX V#(I, N + 1) - .5 * (P#(I, N + 1) + Q#(I, N + 1)) NEXT I 'CALCULATION OF EPOXY RESIN CONCENTRATION FOR I - 1 TO INDEX Z#(I, N + 1) - 21#(I, 0) - K1# * DELT# * Z#(I, N) * V#(I, N) - K2# * DELT# * Z#(I, N) * DM#(I, N) X#(0, N) - (Zl#(I, O) - Z#(1, N+1)) / Zl#(I, 0) IF X#(0, N+1) < 0.7071 THEN GOTO 400 IF X#(0, N+1) - 0.7071 THEN GOTO 380 IF X#(0, N+1) > 0.7071 THEN GOTO 380 380 PRINT "GELATION" 400 NEXT I 'CALCULATION OF SECONDARY AMINE CONCENTRATION FOR I - 1 TO INDEX 001# - DOA# * 108/490 81# - 001# * DELT# / (DELX# ‘ 2) DM1#(I, N + 1) - (DM#(I, N) + B# * EXP(A * V#(I, N)) * (DM1#(I - 1, N + 1) - DM#(I, N) + DM#(I + 1, N)) + B# * A * EXP(A * V#(I, N)) * (DM#(I, N) * DM#(I, N) - 2 * DM#(I + 1, N) * DM#(I, N) + DM#(I + 1, N) * DM#(I + 1, N)) + K1# * DELT# * V#(I, N) * Z#(I, N) - K2# * DELT# * Z#(I, N) * DM#(I, N)) / (1 + B# * EXP(A * V#(I, N))) NEXT I DM1#(INDEX, N + 1) - (DM#(INDEX, N) + B# * EXP(A * V#(INDEX, N)) * (DM1#(INDEX - 1, N + 1) - DM#(INDEX, N) + DM#(INDEX - 1, N)) + B# * A * EXP(A * V#(INDEX, N)) * 144 (DM#(INDEX, N) * DM#(INDEX, N) - 2 * DM#(INDEX - 1, N) * DM#(INDEX, N) + DM#(INDEX - 1, N) * DM#(INDEX - 1, N)) + K1# * DELT# * V#(INDEX, N) * Z#(INDEX, N) - K2# * DELT# * Z#(INDEX, N) * DM#(INDEX, N)) / (1 + B# * EXP(A * V#(INDEX, N))) DM2#(INDEX, N + 1) - (DM#(INDEX, N) + B# * EXP(A * V#(INDEX, N)) * (DM#(INDEX - 1, N) - DM#(INDEX, N) + DM2#(INDEX - 1, N)) + B# * A * EXP(A * V#(INDEX, N)) * (DM#(INDEX, N) * DM#(INDEX, N) - 2 * DM#(INDEX - 1, N) * DM#(INDEX, N) + DM#(INDEX - 1, N) * DM#(INDEX - 1, N)) + K1# * DELT# * V#(INDEX, N) * Z#(INDEX, N) - K2# * DELT# * Z#(INDEX, N) * DM#(INDEX, N)) / (1 + B# * EXP(A * V#(INDEX, N))) FOR I - 1 TO INDEX - 1 DM2#(INDEX - I, N + 1) - (DM#(INDEX - I, N) + B# * EXP(A * V#(INDEX - I, N)) * (DM#(INDEX - I - 1, N) - DM#(INDEX - I, N) + DM2#(INDEX - I + 1, N + 1)) + B# * A * EXP(A * V#(INDEX - I, N)) * (DM#(INDEX - I, N) * DM#(INDEX - I, N) - 2 * DM#(INDEX - I + 1, N) * DM#(INDEX - I, N) + DM#(INDEX - I + 1, N) * DM#(INDEX - I + 1, N)) + X1# * DELT# * V#(INDEX - I, N) * Z#(INDEX - I, N) - K2# * DELT# * Z#(INDEX - I, N) * DM#(INDEX -I, N)) / (1 + B# * EXP(A * V#(INDEX - I, N))) NEXT I FOR I - 1 To INDEX DM#(I, N+1) - 0.5 * (DM1#(I, N + 1) + DM2#(I, N +1)) NEXT I 'CALCULATION OF TERTIARY AMINE CONCENTRATION FOR I - 1 TO INDEX DOZ# - DOA# * 108/872 BZ# - 002# * DELT# / (DELX# ‘ 2) MDM1#(I, N + 1) - (MDM#(I, N) + 82# * EXP(A * V#(I, N)) * (MDM1#(I - 1, N + 1) - MDM#(I, N) + MDM#(I + 1, N)) + 82# * A * EXP(A * V#(I, N)) * (MDM#(I, N) * MDM#(I, N) - 2 * MDM#(I + 1, N) * MDM#(I, N) + MDM#(I + 1, N) * MDM#(I + 1, N)) + K1# * DELT# * V#(I, N) * Z#(I, N) + K2# * DELT# * Z#(I, N) * DM#(I, N)) / (1 + BZ# * EXP(A * V#(I, N))) NEXT I NDM1#(INDEX, N + 1) - (MDM#(INDEX, N) + 82# * EXP(A * V#(INDEX, N)) * (MDM1#(INDEX - 1, N + 1) - MDM#(INDEX, N) + MDM#(INDEX - 1, N)) + 82# * A * EXP(A * V#(INDEX, N)) * (MDM#(INDEX, N) * MDM#(INDEX, N) - 2 * MDM#(INDEX - 1, N) * MDM#(INDEX, N) + MDM#(INDEX - 1, N) * NDM#(INDEX - 1, N)) + K1# * DELT# * V#(INDEX, N) * Z#(INDEX, N) + K2# * DELT# * Z#(INDEX, N) * DM#(INDEX, N)) / (1 + 82# * EXP(A * V#(INDEX, N))) MDM2#(INDEX, N + 1) - (MDM#(INDEX, N) + 82# * EXP(A * V#(INDEX, N)) * (MDM#(INDEX - 1, N) - MDM#(INDEX, N) + MDM2#(INDEX - 1, N)) + 82# * A * EXP(A * V#(INDEX, 145 N)) * (MDM#(INDEX, N) * MDM#(INDEX, N) - 2 * MDM#(INDEX - 1, N) * MDM#(INDEX, N) + MDM#(INDEX - 1, N) * MDM#(INDEX - 1, N)) + K1# * DELT# * V#(INDEX, N) * Z#(INDEX, N) + K2# * DELT# * Z#(INDEX, N) * DM#(INDEX, N)) / (1 + B2# * EXP(A * V#(INDEX, N))) FOR I - 1 TO INDEX - 1 MDM2#(INDEX - I, N + 1) - (MDM#(INDEX - I, N) + BZ# * EXP(A * V#(INDEX - I, N)) * (MDM#(INDEX - I - 1, N) - MDM#(INDEX - I, N) + MDM2#(INDEX - I + 1, N + 1)) + 32# * A * EXP(A * V#(INDEX - I, N)) * (MDM#(INDEX - I, N) * MDM#(INDEX - I, N) - 2 * MDM#(INDEX - I + 1, N) * MDM#(INDEX - I, N) + MDM#(INDEX - I + 1, N) * MDM#(INDEX - I + 1, N)) + K1# * DELT# * V#(INDEX - I, N) * Z#(INDEX - I, N) + K2# * DELT# * Z#(INDEX - I, N) * DM#(INDEX - I, N)) / (1 + BZ# * EXP(A * V#(INDEX - I, N))) NEXT I FOR I - 1 TO INDEX MDM#(I, N+1) - 0.5 * (MDM1#(I, N + 1) + MDM2#(I, N +1)) NEXT I 'OUTPUT INSTRUCTIONS COUNT - COUNT + 1 PRINT A * DELX#; "METER", B * DELX#; "METER" FOR N - 1 T0 MARX PRINT V#(B, N), X#(0, N) WRITE #1, X#(0, N) NEXT N GOSUB 900 PRINT "COUNT- "; COUNT * DELT# * 10; "SEC" NEXT J CLOSE #1 END 900 FOR I - 0 TO INDEX V#(I, 0) - V#(I, MARX) P#(I, 0) - P#(I, MARK) Q#(I. 0) - Q#(I. MARK) Z#(I, 0) - Z#(I, MARX) DM#(I, 0) - DM#(I, MARK) DM1#(I, 0) - DM1#(I, MARX) DM2#(I, o) - DM2#(I, MARX) MDM#(I, 0) - MDM#(I, MARX) MDM1#(I, 0) - MDM1#(I, MARX) MDM2#(I, 0) - MDM2#(I, MARX) X#(I,0) - X#(I, MARX) NEXT I RETURN 10. 11. 12. 13. 14. 15. 16. 146 REFERENCES U.S. Patent #4,752,428, issued to Air Products and Chemicals Inc., June 21, 1988. U.S. Patent #4,77l.110, issued to Air Products and Chemicals Inc., September 13, 1988. Hayes, L.J. and D.D. Dixon, "Journal of Applied Polymer Science", 22, 1007 (1978). Koros, W.J., V.T. Stannett, and H.B. Hopfenberg, "Polymer Science and Engineering", 22, 738 (1982). Elman, J.F., L.J. Gerenser, K.E. Goppert-Berarducci, and J.M. Pochan, "Macromolecules", 23, 3922 (1990). Grimm, H.J. and E.L. Thomas, "Polymer", 26, 38 (1985). Sikorski, R.T. and E. Czerwinska, "Polymer", 25, 1371 (1984). Gilbert, Everett, Sulfonation and Related Reactions, Interscience Publishers, New York, 1965. Private communication with Bill Walles. Wade, L.G. Jr., Organic Chemistry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1987. Walles, W.E. in EQZIiQI Polymers and Strucrures, American Chemical Society, 1990. Ihata, Jyoji, "Journal of Polymer Science: Part A: Polymer Chemistry", 26, 167 (1988). Wang, J., A. Khokhlov, D. Peiffer, and B. Chu, "Macromolecules", 25, 2566 (1992). Hara, M., J. Wu, and A. Lee, "Macromolecules", 21, 2214 (1988). ~ Lantman, C.W., W.J. MacKnight, J.S. Higgins, D.G. Peiffer, S.K. Sinha, and R.D. Lundberg, "Macromolecules", 21, 1339 (1988). Weiss, R.A. and X. Lu, ANTEC '92, 1424. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30I 31. 147 Armstrong, G.R., M.R. Dargin, and L. Johnson, ANTEC ’92, 1904. Crank, J., Tne Mathematics of Diffusion, Clarendon Press, Oxford, England, 1975. Geankopolis, Christie, Mass Transport Phenomena, Holt, Rinehart and Winston, New York, 1972. Hornbeck, Robert W., Numerigal Methods, Quantum Publishers, New York, 1975. Carnahan, Brice, H.A. Luther, and James O. Wilkes, Applied Numerical Methods, John Wiley and Sons, N.Y., 1969. Coalition Technologies Ltd., Freeland, Michigan. Cerfontain, Hans, Megnnnistic Aspects in Aromatic Snlfonarion and Desulfonation, Interscience Publishers, New York, 1968. Skoog, Douglas A., Eringiples pf Instrnmenrnl Analysis, 3rd edition, Saunders College Publishing, Philadelphia, Pa., 1985. Walls, J.M. (Ed.), Methods Of Surface Analysis - Tepnnignes and Applications, Cambridge University Press, New York, 1989. Kalantar, Javad, PhD thesis, Michigan State University, 1991. Klomparens, K.L., S.L. Flegler and G.R. Hooper, "Procedures for Transmission and Scanning Electron Microscopy for Biological and Medical Science", LADD Research Industries, Burlington, VT, 1986. Weast, Robert C. (Ed.), CRC nnndboog of cngnistry nnd Pnysics, 65th edition, CRC Press, Boca Raton, FL, 1985. Gupta, V.B., L.T. Drzal, C. Y-C. Lee, and M.J. Rich, "Polymer Engineering and Science", 25, 812 (1985). Rao, Venkatesh, PhD thesis, Michigan State University, 1991. Kenny, J.M., A. Apicella, and L. Nicolais, "Polymer Engineering and Science", 29, 973 (1989). "7111111111711111711“