SW33?" GE 309mm RAH? Q? POLYMERIC FELMS WWH A VIEW 1% ©MLGPMENT GE SEMULATEQN PROCENRE FOE. CQNTRQL OF HfiAT SEALEMG CYCLE Thesis for {The Deqma' mg Ma S. MICHEGM STATE WNERSITY ‘5 avid P‘egaz 1366 P. v.3 R& An Ema mm. Lm xxxgxwguxxgmgwm 104 mwflxxulugxmxwum w .m U ABSTRACT STUDY OF BONDING RATE OF POLRHERIC FILMS WITH A.'IEW'TO’DEVELOPHIIT OP SIMULATION PROCEDURE FOR CONTROL OF HEN! SEALING CYCLE by David Pagan Ibo lack of direct relationship between the heated bar temperature, dwell time and pressure used traditionally in heat teal strength prediction and control has been realized and in order to overcome it. application of heat transfer theory to com- putation of bonding interface temperatures has been attempted in the past. Thia etudy includes detailed investigation of the deviations between the temperatures computed under several Initial and Bound. cry value Probleniuodcls and the experimentally obtained tempera- tures, identifying the causes for these deviations and determining a suitable model which can be used for the computations and whose rasulta are independent of the heat sealer construction. Several theories of bonding mechanisms in heat sealing are reviavad and on the basis of. these. variables such as viacoeity, fluidity and diffusivity ratio are computed in addition to tempera- ture at the bonding interface. A large number of bond strength determinations of heat seals, made under varying conditions, are tabulated and their plot results in a act of indifference curves believed to be characteristic of the material used. Within the range studied, the effects of pressure increase on the bonding rate are observed with the conclusion that pres- sure in clears of approximately 1.6 p.a.i. slows the bonding pro. case. the indifference curves show clearly the transient nature of the bonding process proving the fallacy of associating a given temperature with a fixed level of bond strength. Finally, a simulation.procedure is preposed, by means of which control of heat sealing conditions is possible and beat seal strength resulting from such conditions is predictable. sum or momma RATE or rommarc was mm ‘A use ro swam or SIMULATION PROCEDURE FOR COHTROL OF HEAT SEALING CYCLE By David Pegas A.THEBIB Submitted to Michigan State University in partial fulfillment of the requirements for the degree of ‘MASTER OF scrzucs Department of Forest Products School of Packaging 1966 ACKNWLEDGEMENTS The writer wishes to express his sincere appreciation for the help and guidance given in the pursuit of this study by Dr. Hugh 8. Lockhart. and for the valuable advice given by Dr. James H. Goff and Mr. Howard C. Blake III. all members of the Faculty of Michigan State University's School of Packaging. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . LIST OF TABLE . . . c . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . . . . INTRODUCTION . . . . . . . . . . . . . . . . . . BACKGROUND . . . . . . . . . . . . . . . . . . . THEORETICAL Adhesion 0 Diffusion Diffusion . . . . . . . . . . . . . . . . . Vi'COBity e e e a e e e a a a e a a a e a e HeatCODdUCtimeaeeeeeeaeaeaa Initial and Boundary Conditions . . . . . . EXPERIMENIAL Tharmistor Calibration . . . . . . . . . . . Comparison of Computed Temperature Functions Heat Seal Bond Strength Determination . . . ANAYLSIB AND CONCLUSIONS Simulation Model . . . Contact Resistance . . . . . . . . . . . . . EffeCt Of Pressura e e e a e e e a c a e a e Balding Rate a e a e e O a e e e e e e a e a Recommendations for Future Study LIST OF REFERENCES . . . . . : . . . . . . . . . APPENDIX I ‘Nbdels . . . . . . . . . . . . . . . APPENDIX II Tables . . . . . . . . . . . . . . . APPENDIX III Derivation of Model III . . . . . . iii . 0 O O . Page ii iv 15 21 24 27 29 32 35 48 63 63 66 72 74 77 86 97 4. 5. 6. 7. 10. 11. 12. 13. LIST OF TABLES Computed Aluminum Resistivity as a Function of Temperature.................... Thermistor Temperature Calibration Data for varying Loading Levels at Sensitivity Setting of .02 Volts per Division . . . . . . . . . . . . . . . . . . . Temperatures Determined Experimentally . . . . . . . . Temperatures Computed by Equation 1 . . . . . . . . . Temperatures Canputed by Equation 2 . . . . . . . . . Temperatures Computed by Equation 3 . . . . . . . . . Temperature Computed by Equation 6 with Finite Thickness of Insulation . . . . . . . . . . . . . . Temperature Computed by Equation 4 with Finite Thickness of Insulation and .001 Inch Teflon Coating . . . . . . . . . . . . . . . . . . . . . Bond Strength in Pounds Per Inch Tabulated for each Heat sealing Time and Pressure Combination and varying Heated Bar Temperature Th . . . . . ... . . Temperatures at Bonding Interface Computed for .002 Inch PolyprOpylene Film with‘varying Heated Bar Temperatures'rh.............oo... Viscosities at Bonding Interface of .002 Inch Poly- propylene Film Computed with Varying Heated Bar TemperaturesTh.o................ Pluidity at Bonding Interface of .002 Inch Poly- propylene Film Computed with Varying Heated Bar Imperatures Th 0 O O O O O O O O I O O C O Q C O O ‘Diffusivity Ratio D/Do at Bonding Interface of .002 Inch Polypropylene Film Computed for varying Heated Bar Temperatures Th . . . . . . . . . . . . iv Page 86 87 88 88 89 89 9O 90 91 94 95 95 LIST OF FIGURES Figure Page 1. Schematics for Derivation of Fick's Lau . . ._ . . . . 22 2. PolymerSidegroupsva.Tg 27 3. Plot of Resistivity vs. Temperature (Computed) . . . . 34 6. Plot of Thermistor Calibration Data . . . . . . . . . 34 5. Temperature Function Obtained By Use of a Thermistor . 36 6. Temperature Function Obtained By Computation Under Mlteeeeesoeeeeeeeeeeeeeae 39 7. Temperature Function Obtained By Computation Under ‘nmllleeeOOleeeeeeeeeeeeesa 40 8. Temperature Function Obtained By Computation Under MOIIIIeeeeseeeeeeeseeeeeeso 43 9. Tuperatura Functims Under Assumption of Semi- infinite Solid (curve A) vs. Finite Thickness of Insulating Material (curve B) . . . . . . . . . 45 10. Temperature Functions Cmputed by Model IV for Contact Resistance Corresponding to .l, .2 and .4K11AirLayet................. 47 ll. Temperature Functions Computed Under Assumption of Haul Surface at Th (curve A) vs. Metal Coated with 1 Nil Teflon and T l/8 Inch Away from “:31 surface (curve B) O O O O .. O C O C I O U C O 49 12. Experimental Heat Sealing Set-up . . . . . . . . . . 51 13. General View of Bond Strength Testing Apparatus . . . 53 15a LightVEIEht Film Gripfi in Use a e e s e e e e e e e s 54 15. Bond Failure Record, Curve A - Peeling, Curve 3 - Break 0 O O I O O O O O O O O O O O O O O O O O O 56 15a. Stress - Strain Curve of 2 Mil PolyprOpylene . . . . . 56 16. Tenerature Functions Resulting from Different Th Value.......................58 V LIST OF FIGURES Figure Page 17. Viscosity Functions Resulting from Different Th '8 1m 0 e e e e s e s e s e o e e e e e e s e 6 0 18. Fluidity Functions Resulting from'Different Th va 1%. O I 0 O I O O O O O D O O O O I O O O O 6 1 19. Diffusivity Ratio D/Do Functions Resulting from DifferentThValues eeeeeeeeeeeeeee 62 20. Bond Strength vs. Dwell Time for TWO Th values and Different Pressure. e e e e s e e e e e e s e e e 65 21. Bond Strength vs. Dwell Time for Different Th values. 67 22. Schematic Representation of Heat Sealing Assembly forHOdelleeseoeesseeeseessss 77 23. Schematic Representation of Heat Sealing Assembly formdelxl eeaesseoesoessesso 79 24. Schematic Representation of Heat Sealing Assembly forMOdEIIIII seeesseeeaaaeesse 80 25. Schematic Representation of Heat Sealing Assembly fordeJ-Iv eooeeeeeeeeeeeeeee 82 26. nodal Scheme for Internal Nodes . . . . . . . . . . . 83 27. nodal Scheme for Interface Nodes . . . . . . . . . . 83 28. Schematic Representation of Composite Solid . . . . . 97 vi INTRODUCTION Heat sealing, a process by which polymeric materials are bonded by application of heat and pressure, is used extensively in a large variety of fabricating methods in the manufacture of widely dif- ferent products, from raincoets to packages. There is hardly any package made from flexible packaging materials in which heat sealing has not been used. In spite of this wide use and the success with which this technique has been employed, the exact nature of the heat seal bond formation is not completely understood. Suppliers and manufacturers of polymeric films usually report material heat sealing temperature ranges of 30 to 150 degrees I, depending on the type of heatsealer employed, where the reported temperatures refer to the hot bar of the heatsealer. Since the temperatures of the film at the bonding inter- face are lower than either the lower or the upper limits of the heat sealing temperature range reported, the useful information obtained from such a specified range is rather limited and reduces to the following: a. The lowest temperature at which heat scaling is possible is lower than the low level of the reported heat sealing temperature range. b. The highest temperature, a heatsealer heated bar can have before degradation of the film occurs, corresponds to the upper level of the range. In order to gain more information on the film's heat scaling properties, users and researchers have tabulated heatsealer tempera- tures, pressures and dwell times against heat seal strength values in order to determine the best heat sealing conditions. The main disadvantage of these results is that they are valid only for the experimental heat sealing system. No conclusive information can be obtained from them for general application. An.improvement in the approach to determination of the best heat sealing conditions was achieved by the use of heat transfer theory, which enables the computation of the bonding interface tem- perature, so making the relationship of the tabulated values of heat seal strength to the temperature levels more meaningful. The validity of the computed temperatures depends heavily on the degree of analogy between the physical model, the heatsealer and the film, and the mathematical model used in representing it in the computational scheme. In the language of mathematicians the compuo tational scheme is referred to as a Boundary Value Problem whose accuracy in representation of the real case depends on.the propriety of assumptions describing the Initial and Boundary conditions of the real system. .A recent deve10pment of surface temperature measuring made it possible to investigate the agreement betweenths bonding interface temperatures, calculated by means of different computational models, and experimentally determined temperature. In this project the writer attempted, after investigation of the reliability of the above mentioned surface temperature measuring method. to determine the most suitable mathematical model possess- ing the necessary characteristics of adaptability so that a wide variety of heatssalers with different physical constructions could be accommodated. Further. various proposed bonding mechanisms of polymers were surveyed, with a view to possible application to heat sealing. Borrowing from studies conducted on the phenomena of ad«. hesion and autohesion of high polymers. the mechanism of diffusion was found to be most applicable. Viscosity, fluidity and diffusivity were added to temperature as the variables of interest and their com- puted values were used in interpretation of heat seal strength re- sults obtained in a series of tests with sealed polyprOpylene. Although the results do not lead to an easily expressible formula. an attempt was made to'construct a procedure by means of which a heat seal strength can be predicted or, if a strength level is predetermined, the conditions at which the seal has to be made can be found. Since the conclusions of this study deal with the strength - . temperature - time relationship, the above procedure can be used in determination of any one of the three variables, given the desired values for the other two. There is some indication of the relationship between host sealing conditions and changes in barrier properties of polymeric films used in packaging, an area into which a further study is presently contem- plated at the Michigan State University School of Packaging. The outcome of the above study together with the results dis- cussed in this paper could be combined into a design evaluation pro- gram for flexible packages by means of which optimal design character- istics could be found using the heat seal strength and the permeabil- ity or gas transmission rate as constraints. The writer believes that charts of bond strength vs. time with temperature, viscosity, fluidity indifference curves should be added by the filn‘manufacturers to other physical and chemical data charac- terizing each particular material and should be made available to every user who would then obtain.useful information more meaningful than the customarily reported heat sealing range. BACKGROUND Throughout the literature dealing with heat sealing, attempts can be found to define formally the phenomenon. The definitions imply their author's Opinion on the nature and mechanism of bond forming. Scarpa (1).. in his paper on joining plastics with ultrasonics, views the surface of the solids to be bonded as having peaks, or asperites, that tower over adjacent valleys by several hundreds to many thousands of atomic diameters. When two such surfaces are put in contact, only the tips of their peaks touch. In ultrasonic bond- ing the rapid and powerful impacts cause a flatening of the surface peaks and, on microscopic level, cause the material to flow to either side of each peak. This increases the contact area and also forces the plastic surfaces tO‘mOVC not only vertically but laterally as well. the friction of this motion generates intense heat at the bonding interface which not only contributes to the tendency of the plastic to flow, so further increasing the area of contact, but also, within a few microns thickness of the interface. stress-relieves the new bonds as they form. These bonds make up the adhesive properties accounted for in general by the Van der Neal’s forces of interatomic or intermolecular attraction, including interaction of polar groups and hydrogen bonding. e Nunbers denote the source as listed in List of References on Page 74 . This explanation places clearly the bonding process into the realm of electrochemical phenanena,‘ where the pressure and best serve only as means of achieving the necessary interatomic and inter- molecular proximity for the forces of attraction to take hold. a different mechanism of bonding is implied by ttuwe (2), who maintains that fusing of thermoplastic polymers is obtained by heating the surfaces of the two layers to be bonded up to a melting temperature while subjecting them to such a pressure that molecular groups near the surfaces flow as into another. lie father points out that this towerature has to be below the decomposition tempera- ture of the material bonded, concluding that the temperature distri- buticn at the bonding interface is of particular importance. Btuwe's approach seems to attribute the formation of a bond to therml motion of the molecular chains, or parts of them, which inter- look across the interface. He does not though, discuss the nature of the forces keeping them in place which may be either rheological only (mechanical) or Van der Wasl’s type forces of attraction. Voyutskii (3), in his paper at the adhesiu and autohesion of polymers, takes a critical look a the theories by which at the pre- sent the adhesion of polymer to polymer can be explained. Making use of experimental results of s umber of Russian researchers, he exam- ined the adsorption, electrical and diffusion theories as related to adhesion of polymrs, noting in particular whether the theories were supported or refuted. The adsorption theory regarding adhesion as a purely surface process explains the strong bond formation as a result of the action of variety of molecular forces. This theory, however, has serious shortcomings. First, while the work of peeling may be as great as 104 ~ 106 erg/cmg, the work required to overcome the molecular forces is only 102 «- 103 erg/m2. Second, the work of peeling an adhesive bond depends on the rate of separation, whereas the work expended on overcoming the molecular forces should not depend on the rate at.which the molecules separate. Third, the adsorption theory cannot explain the good adhesion betwem non-polar polymers. to explain the facts which do not come within the framework of the adsorption theory, an electrical theory of adheeim has been pro- posed (lo). In this theory the adhesive-substrate system is viewed as condmser coated with a double electrical layer formed by the contact of the two substances of different nature. During delanination of the band a potential difference develOps, increasing to a certain limit, until discharge occurs. Although this theory cmintly accounts for the rats of separation dependence of work of peeling, a masher of factors limit its application to the sutusl amnesia of high polymers (5). First, high polymers are able to band together strongly even when there are no electrical effects during lamination. Second, in the case of high-polymers that are dielectrics, it is difficult to seems that electron transfer occurs to any consider- able extent from one polymer to another. Third, the more similar ths nature of the polymers, the greater is their adhesion, while by the above theory, a reverse dependence should have been observed, because the more similar the phases brought into contact, the smaller should the potential difference become . Fourth, if adhesion were determined only by the formation of a double electrical layer, adhesion between polymers filled with car- bon black would be impossible because such mixtures are good conduc- tors. It seems then, that the electrical theory of adhesion is applicable only to polymers which are mutually insoluble. For mutually soluble polymers which are non-polar the electrical mech- anion is impossible and the formation of bond is probably due to interweaving of the molecular chains of the surface layers as a re- sult of their mutual diffusion._ The formation of bond between polar polymers may give rise to electrical layers, but if the molecular chains or their segments are capable of intense thermal motion there will be cross-linking of both layers as a result of diffusion. As the time of contact increases, the effect of diffusion becomes more and more important because of the elimination of the surface of conv- tact and the increase in depth of penetration. The above conclusion has been also reached by Moroccan and Krotove in experimental study of the relative rate of electrical and diffusion processes in the adhesion of polymers (6). The diffusion theory explains adhesion, like autohesion, to consist of diffusion of chain molecules or their segments which form the strong bond between the two layers. Autohesion strength has been found to depend on viscosity of the bonding polymers in which, the lower their viscosity the more easily self-diffusion will occur. Support for viewing autohesion as a diffusion process has been ob- tained by experimentally determined relationships between the auto- hesive strength and the time of contact at constant temperature of polyisobutylene (7, Figure 1). and between the autohesive strength and the temperature with constant contact time (7, Figure 2). In the time-of-contact function the strength at first increases rapidly but slows down tending toward a definite limit. The strength as a function of temperature shows, over the temperature range studied, an exponential relationship. Both the above characteristics seen to he in accord with the diffusion nature of the process. One additional finding reported by'Voyutskii (7) supports the notion.that molecular chain segmts of appruimately equal mgnitude participate in the diffusion process. This finding shows the molecular weight independ- ence of the computed activation energy of autohesion for polyisobuty- ions, which indicates the identical nature of kinetic units partici- pating in diffusion. The diffusion.approach also conveniently accounts for the dif- ference in bonding rates and temperatures of polymera'sith diverse molecular structures. ‘The ease of bonding can be now related to the relative flexibilities of polymer chains which decrease by inclusion of bulky sidegroups. unsaturationa, and polar groups. The same relationship can be observed in the characteristic viacosities of different polymers or in their characteristic glasaotranlition tem- peratures (8). Experimental studies of heat-seal bond strength vera traditionally attempting to view the bond strength as a function of heated bar temper- ature, heat-sealing time and pressure. This resulted in efforts to im- prove the experimental heetsealer design so as to increase the degree of control over the three variables--temperature, time and pressure (9, 10). Results of tests conducted even on these improved laboratory 10 heatsealers could not be, however, correlated either with similar results obtained on another laboratory heatsealer, or with results obtained on actual production equipment. This lack of correlation has been caused mainly by two factors. One, the temperature of the heetsealer heated bar is no indication of the temperature which the bonding interface experiences and which is determined in addition to the temperature gradient (heated bar - initial temperature) by the heat transfer preperties of the total heat sealing assembly. Tao, the heat sealing time, referred to as dwell time, while it repre- sents the length of time for which the outer side of the heat seal: ins assembly is exposed to the heated bar temperature, does not give any information on the temperaturentime function at the bonding interface. The shortcomings which prevented.the meaningful interpretation of heat scalability studies of various materials were realized and the use of heat transfer theory for computation of the bonding inter- face temperatures were proposed by several writers (2, ll. 12). liathematically, the determination of the temperature at the bonding interface involves the solution of the classical heat trans- fer equation formulated on the basis of Fourier's Fundamental Law of Heat Conduction (13). this solution has to satisfy the Initial and Boundary Conditions by means of which the particular model, these solution is being sought. is characterized. the initial and Boundary Conditions influence greatly the computed results which would be only as good as is the accuracy of representation of the actual model, the heat sealing assembly, in terms of mathematically expressed Boundary Conditions. 11 Several mathematical models of heat sealing assemblies are found in literature (2, 11), but to this writer's knowledge, no extensive studies of heatsealability using the computed bond- ing interface temperatures were tmdertaken to this day, partly be- cause of lack of confidence in the accuracy of representation of the actual system and the unavailability of direct measurement method. A recently developed method for measurement of temperatures of thin plastic films (14) has made it possible to compare the tem- perature functions computed under different Boundary Value assumptions with the actually measured one. The determination of the temperature functions under the different models involves considerable amount of computation. hence the use of. a digital conputer was found to be necessary. The models, described in detail in Appendix I. have shown significant deviations in computed solutions from the experimentally determined temperature function, with the exceptim of Model IV whose computed function al- most coincides with the measured me, due to its flexibility of adap- tation with which the physical characteristics of the eXperimental hestsealer have bcm accmodated. It is this flexibility which makes possible the study of tempera- ture . time a heat seal strength relationship independent of the heat- sealer construction and enables the observation of the transient na- ture of temperature, viscosity, fluidity, and on the temperature dependent diffusivity at the bonding interface. For experimental determination of the heat seal bond strength a method was employed, which represents a combination and modification of two test methods used in determination of bond strength of heat seals and adhesives (9, 15) and ASTM method D 882-61T for testing the tensile prOperties of thin plastic sheeting. The eXperimental procedure described in detail under EXPERDENIAL on Page 32, was chosen for two reasons. One, the writer believes that it is suffi- ciently instrumentally controlled to give interlaboratory repro- ducibility. Two, the manner of loading is the same as used in.ASTH method D 832-61T for tensile preperties determination, so allowing the test results to be viewed in terms of the tested material's ten. sile, yielding, and breaking characteristics. The material used throughout the experimental part of this study, polypropylene, was selected because of its relatively high melting point which enabled the testing of the surface temperature measurement method at temperatures up to 250 degrees F, considerably higher than previously tested (14), and because it is considered to be somewhat more difficult to heat seal than polyethylene or PVC. A few links are still missing in the process of application of the diffusion theory to derivation of a quantitative bond strength prediction scheme, either by using a flux diffusing across a surface, or by use of formulas derived by Vasenin (16) on the basis of physio- chemical considerations. This should not hinder the utilization of information obtained in this study for production control or design optimizing procedures. Both of the above mentioned approaches re- quire the knowledge of the number and the depth of penetration of the diffusing molecular chain segments. To the writer's knowledge, no such experimental study with polymers has been undertaken to this time, although, some mthods applicable to polymers have been employed 13 in other material science areas such as metallurgy (1?). One such mthod involves the use of radioactively - labeled molecules with make it possible to observe the rate of diffusion of one component in.o two-component system of uniformychemical composition. The ob» servation could be done by nutcradiogrsphy or any other available radioactivity detection method. The experimentally determined con- centration function could then be used in computation of diffusion flux which should be directly related to bond strength. Even without the availability, at the present, of fibe clearly defined relationship, the results of this study enabled the plotting of a set of temperature . viscosity - fluidity . diffusivity indif- ference curves, which the writer believes to be characteristic of the particular material tested. Since the bonding characteristic of any given material is influenced by any number of factors related to formulation.snd.manner of fabrication, the results of this study could not hold for polypropylene films in general. However, if a family of indifference curves were added to the properties which a film manufacturer makes available to users, and which are specified for each formulation, these could be then used in determining the necessary'heat sealing apparatus settings which yield a desired bond strength level. A relationship between heat sealing temperature, dwell time and loss of barrier properties was observed (18, 19) and a study of it is now being contemplated at the Huchigan State University School of Packaging. Results of this study, provided some definite relationship will be found, would lend themselves to be used in 14 conjunction with the bond strength curves in design Optimizing method by means of which the bond strength - barrier properties affecting variables could be determined, hence the heat sealing apparatus set- tings. Also, given the apparatus settings, the strength and the bar- rier properties of the package could be predicted. TIEORETICAL The following is the discussion of the relevant theories related to the various aspects of this study. Adhesig ~Diffusig Vasenin (16), in his classification of adhesion phenomena by type of bond and mechanism of adhesion, characterizes diffusion as follows: Nature of the forces Intermolecular Magnitude of the forces 10"6 - 10"8 Energy of the bond, erg/bond 10’1“ - 10"16 Mechanism of formation of a bond Interpenetratim of macromolecules and of parts of them as a re- sult of diffusim. Main parameters influencing . the mechanism of adhesion Intersolubility and mobility of the macro- molecules. Mechanism of failure of the adhesion combinatim Extraction of terminal portions of macromole- cules with slight depth of interpenetratim and. rupture of chemical bonds with deeper penetration. Field of applicability lntersoluble polymers in a physical state allowing retention of a certain mo- bility of the molecular Chains 16 On the basis of this characterization, Vasenin discusses the theoretical implications which lead to a quantitative interpretation, all of which are quoted below. "Diffusion Phenomena in Adhesion. 'The question of the part played by diffusion in the phenomena of sticking arose in connection with the in- vestigation of the autohesion of high polymers (2, 22, 25. 26*). It was established that autohesion depends on the time of contact, temperature, the molecular weight, the chemical nature, and the physical state of the poly- ‘mer. The influence of these factors is easily explained by the diffusion theory of sutohesion. Thereafter dif- fusion conceptions were used for explaining the adhesion of high polymers. The mechanism of formation of the ad- hesive bond may be represented as follows (2).' ‘In the contact of two polymers as a result of thermal motion there secure an interpenetration of the chains, governed by the diffusion of the terminal or median 883* ments of the macromolecules. The portions of the sacred- moleculea which have diffused are retained in the poly- more by intermolecular forces. The strength of the ad- hesion combination which is formed is proportional to the number of molecular chains intersecting the interface and to~the depth of penetration of the macromolecules. The latter depends upon the time of contact, the external con- ditions, the chemical nature of the polymers and their physical state. The applicability of diffusion sassptions to real system is determined by the fulfillment of the folv lowing set of conditions (2):. ., (l) the thermodynamic condition: A2801 = (AH801 - Tassel) 4 o the condition of intersolubility of the polymers, and (2) the kinetic conditions.' ’The macromolecules or portions of them must possess sufficient mobility to effect a process of interpenetra- tion. The mobility may be increased by raising the tem- perature or by mixing the polymers with law'nolcculsr weight substances.' vfi—v— fi—‘__ 'e References throughout the quoted text are Vasenin's. 17 'The role of diffusion in adhesion is confirmed by many experimental facts: (1) a connection has been found between the inter- solubility and adhesion of polymers (27, 28); (2) there has been established experimentally the (3) (4) (5) (6) (7) disappearance of the phase boundaries in the contact of mutually soluble components (29): there has been demonstrated an increase in ad- hesion on increasing the duration of contact (30): this being a necessary but not adequate condition for the applicability of diffusion concepts to real adhesive-substrate system. In adhesion to porous bodies and also with the adsorption macho enism of formation of an adhesive bond as a re- sult of decreasing flow and adsorption of poly- mers we also observe an increase in adhesion with increasing time of contact; an increase in adhesion has been established with increasing temperature of contact (30), but its exponential character cannot, of course, be used as an unambiguous proof of the diffusion mech- anism of the formation of the adhesion bond, since the same type of dependence of adhesion on temperature is observed also with adhesion to porous bodies, and with adhesion due to chemi- sorption of an adhesive on a substrate; ‘vith increasing molecular weight of polymer (3) there is a reduction in the number of terminal segments capable of diffusion, therefore with in- creasing molecular weight the adhesion decreases; the rate of diffusion depends upon the magnitude and shape of the diffusing molecules (32); the greater the dimensions of the molecules and the more short side branchings they have the lower the rate; it has been established experimentally (33) that in the adhesion of polybutadiene to ce110phane, increasing the number of butadienc groupings in the l, 2 position reduces the ad~ hesion; sufficiently long side-branchings play the part of terminal segments and increase the adhesion: the flexibility of the macromolecules has an ex- ceptionally great importance in the interdiffusion of polymers (33) but the quantitative influence of this factor on adhesion is very difficult to inves- tigate; it is established for instance that increas- 18 ing the number of styrene groupings in a butadien- styrene cOpolymer reduces the adhesion as a result of the reduction in the flexibility of the chains.‘ 'Quantitativo Interpretation.’ 'A quantitative interpretation of these experi- mental facts is a difficult matter. Its complexity lies in the impossibility of varying only one of the factors which influences adhesion while main- taining the others constant. For instance, in inves- tigation the effect of structural peculiarities of macromolecules on adhesion, one must note that dif- fusion is altered, not only by'the geometrical form, but also by the flexibility of the macromoleculs. In addition, constant molecular weight of the poly- mer must be maintained otherwise this too will affect the adhesion.‘ 'Adhaaion of High Polymers.' 'To predict adhesion quantitatively it is necese sary to solve three problems (34): l) the nature and magnitude of the intermolecular forces govern. ing adhesion: 2) the number of macromolecular chains intersecting the interface in the formation of a bond: 3) the depth n of penetration of macromolecules.’ 'In the first problem we must realize that if the ‘molecules of a high polymer diffuse mutually to a depth corresponding to the length of a segment of a macromolecule with a degree of polymerization below 200-600, then destruction of the bond takes place ‘with sliding of the chains. If the portion which has diffused is greater, then destruction of the bond occurs by chemical bond rupture of the macro- molecules. The critical value of the length n (i.e.. “er, of the molecule in terms of groups of atoms. at which an alteration in the mechanism of failure occurs, is determined by the ratio: fi ' “or ' fch where it is the force of interwolecular interaction of the atom group; fch the force required for rup- ture ofa chemical bond. Than the conditions under which there is affected one or the other mechanism of destruction of contact will be: £1 a n <.fch the mechanism of chain slips (6) f1 . n > {oh the mechanism of rupture of chemical bonda.‘ (7) 19 'Corresponding to conditions (6) and (7) the adhesion strength depends upon the depth of pene- tration only to nor’ The adhesion strength for condition (6) is equal to: P - f1 Q ' n 1 while for condition (7) r -Z fchQ where 2: indicates the need to take into account the interdiffusion of the components.‘ 'To solve the problem of the magnitude of the intermolecular forces it is possible to make use of the theory of intermolecular interaction. For viscous flow we may also make use of the relation- ship determining the friction force for each group of atoms in the form (35): where m is the mass of interacting groups; 1’ the frequency of interaction; V’ the rate of movement of a molecule (the rate of extraction).' ‘The number of molecular chains intersecting the interface may be determined with the provisos that 1) only the terminal segments of the macromolecules diffuse; 2) the polymers are not treated so that the molecules become oriented; 3) the macromolecules are non-regular and give random, non-ordered arrangement of the chains.’ 'In 1 g.mol of any given substance there are N molecules occupying a volume V" . The number of particles in one cu. cm. is N/VlM ; the number of terminal segments in 1 cu. cm. is equal to 2 N/V" while for 1 sq. cm. it is (ZN/V" )1] or Q - (my/u)“ (8) here ,0 and H are the density and molecular weight of the polymer. Equation (8) determined the sta- tistical mean number of terminal segments in 1 sq. cm. As the process proceeds the number of segments intersecting the interface increases by diffusion of the more deeply placed terminal segments and the possible diffusion of middle portions of the chain. The role of the latter in the adhesion of high poly- mers is not yet explained.‘ 20 'The most complex task is the determination of the depth of penetration of the macromolecules. It may be solved on the basis of Fick's second law. A special feature of the problem is the de- pendence of the coefficient of diffusion of a macromolecule upon its length. Since the length of the diffusing portion of a macromolecule in- creases as the adhesion bond is formed, the co- efficient of diffusion also changes with time.‘ “Problems of this type are considered in a sum. ber of text books on diffusion. Making use of standard methods an approximate solution can be found. For the depth of penetration of macro- molecules in terms of groups of atoms, we obtain the equation (34)! $921 one tN/z where l is the length of the 0.0 bond; 0‘ the angle complementary to the valence angle; A? a constant determining the alteration of the co- efficient of diffusion with time; and K9 a constant characterizing the properties of the diffusing molecules and the diffusion medium. In particular, lg depends upon the flexibility of the diffusing macromolecules and the tempera- ture of cmtact.’ (9) 'For condition (6) taking into account (8) and (9) the adhesion strength 21 cos ok/z 2 2/3 [2; £1643 (10) Equation (10) is correct for n 4 n or g 4; , as determined by cr or 2/(1-0) _ Zlcos on; fish tor [TI ‘0 it For condition (7) t >> tcrzz t] ‘(1 ° [”72 r . (2N)2/3 if“ .42” 21 'Sincs for polymers in the viscous and high- elastic states “or is quite large (f1 < < fch) it is possible to make use of (10) over a wide time range. According to this equation the ad- hesion strength is a parabolic function of the time of contact, inversely proportional to the 2/3 powgr of the molecular weight and proportional to K D characterizing the mobility of the molecular 611811195. ' 'The diffusion theory of adhesion is faced with the problems: 1) of direct experimental proof of diffusion of macromolecules or of their portions in adhesion phenomena; of quantitative investigation; 2) of the influence of the molec- elar weight and the part played by the diffusion of the middle portion of the macromolecules in the adhesion of polymers; 3) of the temperature dependence of the adhesion for the determination of the energy of activation of the process and its comparison with the energy of activation of the diffusion processes in polymers; 4) of the dependence of adhesion on the chemical nature of the adhesive and the substrate (36).'" i fusi Diffusion is a movamnt of molecules or stone under the influence of a concentration gradient. leading to an equalization of concentra- tion within a single phase (20). II macromolecules. diffusion is closely related to Brownian motion and it could be said that the molecules or particles of a sub- stance diffuse because of their Brownian motion (21). Transfer of heat by conduction is also due to random molecular motions, and there is an obvious analogy between the two processes. This was recognized by Pick. who first put diffusion on a quantita- tive basis by adopting the mathematical equation of heat conduction derived earlier by Fourier. The mathematical theory of diffusion in isotropic substances is therefore based on the hypothesis that the rate of transfer of diffusing substances through unit area of a upw- 22 section is preportional to the concentration gradient measured normal to the section, i.e. gg_ F - - D dx where F a rate of transfer per unit area of section C - concentration of diffusing substance x - the space coordinate measured normal to the section D diffusion coefficient. In diffusion involving dilute solutions D can be considered as constant, but in diffusion in high polymers, it depends very markedly on concentration (17). For unsteadyostate conditions there will be an accumulation of diffusible matter in a unit volume, and the concentration at any point within a solid will vary with time, hence gg'fl O. The in- crease in the amount of substance within a volume element bounded by two parallel planes P1 and P2, having a unit area and located at the distances 1 and x-+ dz, will be equal to the difference in the flux Pr and Px-+ dx shown in Figure l. i F F | i c 'L n f I x x-+ dx Fx dF F + dx = F + __x.dx X x dx Figure l Schematics for Derivation of Fick's Law Thus, the flux at x-+ dx is dF a. 9-C— .—d—. dc 23 and at x is dc F: a ' D dx The difference between the two is figs-.1123? dz dx dx . de Since 15E' is equal to the negative rate of concentration change, 1.80. 9.25.39. dx dt we obtain the Second Ficg's law g9, d 5 dt 'Ddx Where D is assumed to be constant at constant temperature. The diffusion coefficient is a measure of the ease with which molecules move within solid. we know, however, that the movement of molecules or their segments within high-polymer is related di- rectly to the thermal agitation, hence the diffusion coefficient is Strongly temperature dependent and varies with temperature accord— ing to the Arrhenius otype equation: O D - Doe fin where D a diffusion coefficient D a a constant (in diffusion of gases taken as the diffusion coefficient at infinitely high temperature) I! I activation energy R - gas constant I - temperature in 9K . 24 Fick's law cannot be directly applied to diffusion of high polymers because of the complex nature of the diffusion process. The diffusion flu: there will probably depend not only on tempera- ture and concentration, but also on the change of concentration gradient and on the nature and extent of the "grain" boundaries (20). Eiscosity A characteristic property which determines the flow of any fluid is its viscosity. Viscosity may be defined as internal friction or as the resistance of a fluid during flow (20). It is more exactly defined as the ratio of the shearing stress to the‘ rate of shearing, i.e., ’7 f § where ’7 I the viscosity 8 . shearing stress D a rate of shearing (22). Viscosities are usually reported in terms of poises. A fluid has a viscosity of l poise if steady tangential force of l dyne pro- duces a relative velocity of 1 cm per sec between two parallel planes of area 1 cm2, separated by a distance of 1 cm and immersed in the fluid. The reciprocal of viscosity is a measure of fluidity which may be loosely viewed as the ease of flow of a substance. Formally de- fined 25 where R a fluidity S a shearing stress D a rate of shearing . It can be seen that R I7;- and if I) is in poise-3,3 is in the. rhe - poise-1 Some liquids may be supercooled to form glasses without crystslization taking place as the temperature is lowered. In such materials the viscosity changes at so great a rate that over a small temperature interval the character of the mterisl changes tron a liquid to a rigid solid or glass. Many other properties of a substance also show a dramatic change in this glass transition region. The nature of glass transition is not completely understood and in the thermodynamic sense should not be called a transition. Polymers have glass transition, and from the effect of these transi- tions it is convenient to treat the phenomenon as a true transition. A polymer above glass transition temperature 1'8 , may exhibit viscous, viscoelastic or elastic behavior, depending on the molec- ular weight or other characteristics related to geometry. Below the '1" , molecular motion is frozen in. At '1‘ the polymer has expanded to the extent that there is enmgh free voltmo available in the material for molecular motion to begin. Molecular segments occasionally have room enough to jump from one position to another with respect to their neighbors at this temperature. At 1'8 the viscosity of a large number of polymers is roughly 101'3 poises. For normal liquids well above 'I.‘g , the viscosity 7 my gen- erally be represented as a function of temperature by 26 7 " ‘70 ° where I70 a constant representing the viscosity at infinite absolute temperature AH . activation energy of viscous flow R II gas constant T n temperature in 9! For high polymers between,‘!é and 100 degrees above I“ , it has been found that the viscosity may be more accurately approxi- mated by 9 -17.44cr-r) [51] (W5- where r and 1:8 are in °r. 7": taken as 1013 poises (23). Glass transition temperature Th , is related closely to chemical structure of a polymer. The most important factors determining the value of It seem to be the flexibility of the polymer chain.snd such chain flexibility related factors as steric hindrance and bulkiness of the side groups attached to the backbone chain. Bulkiness of a side group increases the glass transition temperature '1' 3 be seen on the following illustration (Figure 2) in which the in» ,ascan creasing side group bulkiness is compared with the resulting 1'8 . 27 Figure 2 Polymer Sidegroups vs. '1' 8 Poly??? - S ide group 1'8 (06) Polyethylene ~11 0120 P0 lypr0pylene -CH3 .10, ~18 Polys tyrene .@ 100 , 105 Polyvinylcarbazole «g 208 Has duction when different parts or a body are at different temperatures heat flows from the hotter part to the cooler. The heat transfer occurs'by three possible methods: 1) conduction, 2) convection, and 3) radiation. In solids. convection is absent al- together and radiation negligible, hence conduction is the applicable method. . The basic law which quantitatively defines heat conduction is attributed to Fourier. The one-dimensional form of the Fourier law states that the Quantity of heat dQ donducted in the x-direction of a homogenous solid in time dt is s product of the conducting area A normal to the flow path 3. the temperature gradient g along this path, and a preperty K of the cmducting material known as "thermal conductivity” . i.e. , §%=-M§ (13>. 28 The one-dimensioml heat cmduction equation can be now derived as follows: Assume two parallel planes of. unit area each, within a homogeneous circular rod with associated flow rates per unit time across than £1 and £2. If no heat flow takes place through the curved surface the rate of heat increase in the section between the two planes is (1 0 f2 0 Also, if '1' is the average temperature in the section, d: is the distance between the planes, and P and C the density and specific heat respectively. the rate at which the sectia gains heat ‘3 equal to fodlbg . 10.0; (f . t )dt . “I c "1 2 I“??? Making use of Fourier Law for unit area 6 d1‘ 3% u 0 K3! the rate of gain of heat through the plane at x can be expressed as «a - ~ taétfir-m and through the plane at x + dx. as de—dx - ' Rfig + gt”) dt The total increase in internal energy within the section will then equal . dT d: . (£1 ~ f2)dt - ,aceca-Edt 29 and de a de+dx‘+ d! Substituting the already derived expression for dQ‘ and an+dx . we obtain: «(‘12: d d d! 32M: -- -x(;2-)dc «(gums + pCdX‘a'Edt . Rearranging and cancelling we obtain the final relationship: 2 d.T .. d! K?"’ 0“ s dxz :0 dt Assuming that for the temperature interval of interest the conductivity. density. and.heat capacity of a material stay con- stant, we group the constants and define s new one called thermal difEMIVtty, ices . .5...“ IPC Hence, to determine the temperature within s solid st point x and time t , a solution has to be {and which satisfies the 9.;- dimensional Heat Conduction Eguatigg as follows: dzr . A“ s :3 «a i ial Bounds Candi Before a particular solution of conduction can be found, it is necessary to determine the formulae which will express the Initial and Boundary Conditions which the temperature satisfies. These are 30 the mathematical statements of hypotheses founded on the asperimentalo 1y or otherwise obtained knowledge of the characteristics of the sys- tem under consideration. Assumptions that in the interior of a solid temperature is a continuous function of distance and time. and that the same holds for first and second differential coefficients with regard to dis- tance and for the first differential coefficient with regard to time. do not hold at the boundary of a solid and at the instant at which flow of heat is supposed to start. Initial and Boundary Conditions arising in the mathematical theory of Conduction of East are as follows: a. Initial conditions. The temperature throughout the body is supposed given arbitrary at the time coordinate ears. I - f(x) at t - 0 b. Prescribed surface temperature. This temperature may be constant. or a function of time, or position, or both. For example, !‘- t. at t - 0 and a n O . c. No flux across the surface. The surface is perfectly insulated. fi-o. d. Linear heat transfer at the surface. The "radiation” boundary conditions. If the flux across the surface is proportional to the temperature difference between the surface and the surrounding medium, so that it is given by Ila-1'0) 31 where To is the temperature of the medium and H is a constant. the boundary condition is then 1.5+... .10, .0 or g, +h('l’-1’°)==O,whereh= dx Null as it approaches care this tends to condition "c" and as h approaches infinity it tends to condition "b" . ‘ The quantity II has been called the "Outer" or "Surface" Conductivity. It is often simpler to specify thi surface thermal resistance per unit “8‘. n . i C e. The surface of separation of two media of different conduct- nuns. r1 . and IL2 . Let 11 and '12 denote the tanperatures in the two media. Then the flux over the surface of separa- tion is x “1 . 6!2 In“ ‘27:: ' and if at the surface of separation the temperatures of the two media are the same we have also 1‘1 “ :2 This assumption will be valid only for solids in intimate contact. In other cases the rate of transfer between the two surfaces may be proporo tional to their temperature difference, so that or «1-5;;- = sci-1 ~ 1'2) . (24). The above list is by no means exhaustive and the satiation used has been adjusted to Oneodimensional problems. EXPERIMENTAL The following is s description of all experimental phases of this study arranged in the order in.which they were made. ghermisto; Qalibragigg The method and equipment have been described in detail in (ls), and this particular study was undertaken mainly to extend the tem- perature interval of measurement up to 2509?. The thermistor was a one nil metalized Hercules Powder Pro- Fax polypropylene bi-axially oriented film, out to eight inches by one inch on a Model CDC 25 Precision Sample Cutter made by Thwing - Albert Instrument Company. The ability to measure surface temperature by the aluminized film thermistor rests on the temperature dependence of specific resistance of the thin aluminum layer which has to be sufficiently linear in order to enable interpretation of the voltage function detected on the oscilloscope screen. Kahlbaum (25), reports the constants for pure aluminum to be used in the temperature-resistivity relationship. rt - r; [1+ o<(t-t')10'3+ B(t«t')210‘6] where rt for a solid - resistance at temperature t of a specimen which at tn’has s length of 1 cm, and a uniform transverse sectional area of 1 cm2. 32 33 for t 6 t ‘ t 1 2 t. - 0°C 3‘ 1s80 r; - 2.62x10‘6 ohm-cm. t1 - ~so°b O‘ on O O O l l I 120 100 01 m1]. .2 mil .4 mil I l I l I T .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 Time in sec Figure 10 Temperature Functions Computed byiModel IV for Contact Resistance Corresponding to .1, .2 and .4 Mil Air Layer 1 l T 2.0 48 The good agreement between the computed and measured functions over the temperature range studied, 80 . ZSCOF, seems to support the conviction that,with enough care taken in translating the physical features of the heat sealing assembly into the computational para- meters,the Beck's program can be used for simulation of the thermal changes at the bonding interface in heat scaling. The importance of taking into consideration all the physical features of a heat sealing assembly is best illustrated by observing the differences in functions generated under the assumption, 1) metal surface of the heated bar has the temperature T5, and 2) the metal is coated with .001 inch thick Teflon and Th is the temperature at the thermocouple located 1/8 of an inch away from the surface (Figure 11) 0 Heat Seal Bond Strength Determination The following is a description of the procedure employed for compilation of bond strength values of heat seals prepared under varying conditions of temperature, time,and pressure to be studied with respect to the computationally detemmined bonding interface temperature and with respect to viscosity, fluidity, and diffusivity ratio which depend on temperature. goat Sealer - Laboratory heat sealer modeled after Olin.Mathieson Chemical Corporation design (10), used with a control consol equip- ped with a temperature controller, electric timer and a pressure gage. Tempera tur e 0F 49 240 _ 220 ‘4 200 '- 180 - 160 "‘ 140 120 ‘ 100 " I F I I I I I I I .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 Time in sec Figure 11 Temperature Functions Computed Under Assumption of Metal Surface at Th (curve A) vs. Metal Coated With 1 Mil Teflon and Th is 1/8 in.Away From Metal Surface (curve B) 50 The heat sealer is in the form of a table 835 inches by 4 inches, with an Opening at the center to admit the heated bar, located at a position below the sealer plane. An air piston moves the bar to a sealing position k inch above the plane. The sealing bar is made of aluminum with a surface coating of approximately .001 inch thick Teflon. The temperature sensing element, thermocouple, is imbeded in the metal bar at an 1/8 inch distance from its surface, and the bar is heated by means of a 150 watt heating element mutated in its lower part. The sealing surface has the dimensions of one inch width and 635 inches length. The sealing pressure is determined by dead weight emsisting of an aluminum base plate 5 inches by 63 inches wide and long respec- tively, weighing 21.94 was Q Which lead plates, each weighing approximately one pound, can be added. The sealing time, dwell time, is cmtrolled by a timer which is started by means of a microswitoh adjusted so that the timer is connected at the instant at which the sealing bar first touches the film. the timer, at the and d the pro-set interval, actuates a solenoid valve which releases the air pressure to the air piston, causing the sealing bar to retract at the end of the sealing cycle. the thermal insulatia used was a flat piece of menial neoprene l/l6 inch thick, 4 inches by 635 inches. the entire heat sealing set-up is shown in Figure 12. Film Smles . Samples of polypropylene film, MEANS-A made by Avisun Corporation, approximately 2 mil thick, were cut to 3 inches 51 an: new wGHHmom umom Hmuaoefiuomxm NH shaman 52 by 9 inches. Each sample was folded in half and heat sealed by placing it across the heatsealer table in such a manner that a one inch wide heat seal was made approximately one inch away from the folded edge. This heat sealed sample was then cut to testing dimen- sions, one inch wide and five inches long, with the inner edge of the seal, fin-type, located exactly in the center. Icnsile 3::er Easter «- For bad strength determination, a Baldwin - Emery 58-4, Testing Machine, Model PCT, made by Alsldwin-mLima-Hmilton Corporation, was used. As a sensor, a Load Cell Model LBZTC‘IIO-SSO, with l" 10 pounds range, made by Statham Instruments Corporation, was mounted on the stationary beam of the testing machine. The transient features of the heat seal failure were recorded on a Baldwin Stress-Strain Recorder, Model MA 13. .A general view of the testing equipment is shown in Figure 13. In order not to limit excessively the effective range of the load cell, a special set of film grips made of light~weight materials, aluminum and magnesium, were constructed. “to prevent snarl-as. the gripping faces were coated by a layer of polyvinyl «cuts, Elmer's Glue, approthately 1/48 inch thick. .A closeuup illustration of the grips in actual use is shown in Figure 14. Procedure - Ten heat seal test samples were made for each combination of heated bar temperature, pressure and dwell time listed below: Heated bar temperature Th-°F: 300, 315, 320, 325, 337.5 Pressure levels . p.s.i.: .68, 1.15, 1.61, 2.09, 2.55, 3.47, 3.94 Dwell times - sec: .2, .5, 1.0, 1.5, 2.0, 3.0, 4.0 53 osuouonn< woauooa nuwoouum coon mo Boq> sundown ma ouawqm 54 Figure 14 Lightweight Film Grips in Use 55 The polypropylene film samples were conditioned to 73°F and 50% LR. before heat sealing and allowed to return to these condi-- tions for a period of not less than 30 and not more than 45 minutes between completion of the seal and its actual use in bond strength determination. ‘ For the bond strength determination the sample was clamped into the grips in such a fashion that the load was applied perpendic- ularly to the seal plane, ”lb-test or peel-test. The conditions of testing employed were times specified under Method A of ASTH D 882- 611‘ for Tensile Properties of Thin Plastic Sheeting, i.e., Initial Grip Separation of 2 inches and Rate of Grip Separtion of 20 inches per minute. Ten samples prepared under identical conditions of heat sealing were tested and their results averaged. All the samples were cut in the Machine Direction. The failure of each bond was recorded on the Stress-Strain recorder and the record interpreted as follows: a) Bonds failing by peeling had a characteristic record as sham: in Figure 15, curve A. The bond strength was taken as the mean value of the Stress-Strain curve over the en- tire length of bond separation, i.e., 2 inches. b) Bonds failing by tear or break at the inner edge of the seal had a characteristic record as shown in Figure 15, curve B. The bond strength was taken to be the Breaking Factor of the ASTM D 882-61'1‘ Method, i.e., the maximum force recorded at the point of failure, reported for the width of the sample. Load Load 55 h- Bond Strength Bond Strength _____“_fl__4 Length Figure 15 Bond Failure Record, Curve A - Peeling, Curve B - Break _- __ _Y_igld_Point Second Yield—Faint Length Figure 15a Stress Strain Curve for 2 Mil Polypropylene 57 The bond strength results were identified by their respective heat sealing conditions and are tabulated in Appendix II, Table 9, pages 91-3 . in which the values are the averages of ten tests and are reported for each temperature Th, dwell-time, and pressure com- bination. In order to relate the bond strength results torthe properties of the material used, ten samples of the-film were cut in the Machine Direction and their tensile properties determined under the same con- ditions used for the bond strength tests. The characteristic Stress- Strain curve is shown in Figure 153 and the average values of ten tests were found to be as follows: Film thickness: .00206 inches Sample width: one inch Yield point: 7.76 pounds per inch Second yield point: 5.81 pounds per inch Breaking factor: 11.26 pounds per inch Elongation at break: 5322 The Beck's computational model was then adapted to simulate the heat sealing system used in preparation of the seals tested and the temperature functions at the bonding interface were generated. The computed temperatures appear in Appendix II, Table 10. page 9!», and are plotted in Figure 16. Each computed temperature was used within the same computational model to determine the values of viscosity, fluidity, and the diffus- ivity ratio D/Do. The following formulae were used: o.¢ mosaob n8 acouomman scum mewuaomom mcoHuocnm enououoeaoa ca opnwam use as mafia n.m o.m m.~ o.N m.H o.H m. _ _ h pi _ p _ - ooH ONH oea r.l ooH mooom u H a woman a H .I com eooum u as fi .| CNN woman a H eomgmm a se I can l.om~ l.ow~ I.oom Jo oznneaedma; 59 17.44 ('f-‘l‘ ) “mtg- 1) V1.1. = 71:81:10 3 where 71.1. a viscosity in poises I71. - 1013 poises 8 1‘38 = temperatures in OK 2) R =- 7rd where R =- fluidity in the E 3) 13/120 =- e' RT assuming that DIDo at 1‘8 - .0001 E - 48.08 cal/moleul n - 1.987 cal deg.1 - 2 1'8 63%: The resulting functional values are tabulated in Appendix II, Tables 11, 12. 13, pages 94-5 , and their plotted curves are shown in Figures 17, 18, 19. Viscosity in Poises 35 -‘ 30 -fi 25 -— 20 - Th - 300°F Th a 315°? 0 Th = 320 F 10 -— l' 5 _ ! l l l 1 I I j .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time in see Figure 17 Viscosity Functions Resulting From Different Th Values Fluidity in the x 102 7O 60 50 40 3O 20 10 61 Th - 337.5°F h I 325°F Th - 320°? Th - 315°F r‘ ‘ I *I l I I I .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time in sec _ Figure 18 Fluidity Functions Resulting From Different Th Values D/Do in 7. 62 94.4‘— 94.3-— 94.2- 94.1.— 94.0-- 337.5°F H :I‘ ll Th . 325°F 320°F 315°F 93.9— 93.8 — P3 '3‘ ll 300°? F] 5" I 93.7 «— 93.6 - 93.5 _ 93.4 ._ 93.3 ._ 93.2 .. 93.1 ._ I I I I I I I .5 1.0 1.5 2.0 2.5 3.0 3.5 Time in sec Figure 19 Diffusivity Ratio Functions Resulting From Different Th Values 4.0 ANALYSIS AND CONCLUSIONS Simulation Model - Comparison of the temperature functions (Figures 6. 7. 8, 10) makes obvious the importance of including all the physical features of the heat sealing assmbly in the com- putational method employed in simulation of the transient tempera- ture behaviour at the bonding interface. Since the Beck’s approach enables not only the manipulation of parameters according to the heat sealing systan modeled, but also its solution agrees closely with the experimentally determined temperature functim, it is ideally suited for studies of heat sealing. Its use does away with the effects of. heatsealer construction variations :shich made until now any interlaboratory correlation and reproducibility impossible. ggntact Resistance '1: By the rather indirect method of contact resistance determinaticn, Akers (14) found 0.0002 inch to be an appropriate value for contact resistance for polystyrue. For poly— propylene the appropriate contact resistance was determined to cor- respond to 0.0001 inch. Noting that both the polymers possess the same basis molecular chain structure and vary only in the type and size of the side groups, phenyl and methyl respectively, the contact resistance seems to be related to the bulkiness of these side groups in manner similar to the glass transition temperature. Extension of this line of reasoning leads then {to expectation of a amch lower 63 64 contact resistance to be present in heat sealing of polyethylene which has a Smaller hydrogen in place of methyl and phenyl groups of the above mentioned polymers. Effect of Pressure . In Figure 20 the experimentally determined bond strength resulting from the same temperature settings and vary- ing pressure levels are plotted. Each curve is associated with dif- ferent pressure. It can be observed that initially the bond strength values rise with increase in pressure Which indicates that some amount of pressure is necessary to overcome a kind of "bonding surface re— sistance". This resistance may be related to the thermal contact resistance although other factors are necessarily at play, since the thermal contact resistance did not require an increase in pressure to overcome it, or rather increase in pressure did not eliminate it (Table 2). With further increase in pressure the bond strength values tend to become lower, or the rate of bond strength growth slows down as if the process of bond formation, diffusion, were laboring against some restrictive force. Based on speculation only, this could be viewed as a mass flow of the softened material which, because of resistance offered to it in the perpendicular direction due to higher viscosityin locations closer to the unheated side of the heat sealing assembly, tends to be forced by the pressure to move in parallel to the plane of bonding with the result of lesser contribution to the bond strength assumed here to depend on the number and depth of molecular segments penetrating across the plane of initial separation. The adverse effects of excessive pressure were discussed by several writers and remedial changes in heatsealer die profile were made, which sought Bond Strength in p.s.i. 3.5 3.0 2.5 2.0 1.5 1.0 65 .68 p.s.i. 1.61 p.s.i. 3.94 p.s.i. I F T l l l 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time in see Figure 20 Bond Strength vs. Dwell Time for Two Th and Different Pressures 66 to minimize the formation of head by "extrusion" of molten material from the heat seal area, or to prevent "thinning" of the heat sealed area, in particular, the border line at which the heat sealed and un-heat sealed areas meet. Both the bead and the thinning cause a weakening of the material strength preperties at the edge of the seal. Exactly at this edge failure occurs when the strength of a seal is tested which leads often to the erroneous conclusion that the bond is stronger than the material itself. On the basis of the quite limited amount of experimental evidence obtained during this study, this writer is inclined to»con- clude that pressures in excess of some sdndmel required level are not only unnecessary but also detrimental both to the bonding rate, speed at which a bond is formed, and to the resulting bond strength. ondi - Plotting the bond strength values from Table 9. in Appendix II. for single pressure level and all Th results in a group of curves (Figure 21) from which interesting observations can be made. The bond strength as represented by curve 1 and 2 reaches maximum after approximately 2 seconds and starts slowly to decline ‘with prolonged dwell times. This could be due to two distinct mech- anisms. First, a number of low molecular weight polymer chains in. volved in bond formation in the early stages of the process moves on extended heating completely across the plane of the bonding inter« face, hence eliminating their contribution to the total bond strength. Second, during the initial period of heat sealing, segments and termi- nal portions of the molecular chain move across the plane of separation with the result of stressing some from the surface distant portions of the same molecular chain. Upon extended heating the molecular chain F.jl\tl|v 11 \ ‘ 14...]! 67 momma no sunueuoeaeu announce a nu moomw mo nanoseconds unnumsou I HHH» mommu we shrunkeneeu nonsense 0 HM» enema we ensue—ensue uaeumsoo .. Hp women «0 announces”. announce .. p mocha no senescence» squeegee a so mooem no shamanism.» 3338 s a meomu no unaumuemaeu unmumaou a HH mocha mo unsounonaou announce s H mom.hmm we :8 you eumnuuun menu I m somNn no as not sameness egos o a mdomm no on new newness» neon c m semen so no not assesses ence . N sooom mo ea soc nonsense econ . a asaam> ea uaouoeman use «see Hesse .s> nonsense econ Hm shaman one ca saga o.¢ m.m o.m m.~ a.“ m.H o.H m. p b _ _ _ _ _ rill! H 3‘\ I’ll-I... HHH ill i IIIII . /// a ‘ III, // > “ .II II . N H» “ HH> II —-————‘--——'v 1],]! II ———_ II III II n II II HHH> V\ / \ .II// m [I / n / an / / / / l.o.H l.o.~ l.o.m I o.¢ I.o.n |.o.o “sqI u; qnsuesns puog 68 is allowed to achieve a configuration of least stress, or equilibrium of forces, which may result in total retraction of some, and partial retraction of others, to the bond contributing segments. Each of the curves, 1 through 5, in.Pigure 21 represents the bond strength results obtained for each of the five initial temperatures Th for which we also have the computed temperature, viscosity. fluidity and diffusivity ratio D/Do functions (Figures l6, 17, 18. 19). It is now possible to plot on these bond strength curves the points of common value of the variables which. vhss.connected by s snooth curve, give rise to s set of level curves 3'. since no three-dimmsionel relationship is intended to be implied, indifference curves, each representing a constant twperature. viscosity, fluidity and DIDO. li'he indifference curves are identified on Figure 21 by Rm male I through 1!. The shape of the curves seems to indicate s hyperbolic or exponential relationship betvess.ths variables considered above and the experimentally determined bond strength values. the form these indifference curves could take are as follows: mm ~- I - Act-WM W - x - Ae“““°’+n ': n x.‘ -bx - x t‘a e c +-D 1‘ 2 the shape of the indifference curves clearly shows that the basically static approach of associating temperature directly with bond strength by use of steady state temperature profiles is not ap- plicable. The transient features of. the temperature. viscosity. flu- idity and diffusivity ratio, in particular, the rise rates resulting from the heat sealing sssmbly's initial temperature gradient. are the 69 determining factors in bonding. In other words, the greater the thermal ”shock”, the better or stronger bonds are obtained. For example, if the bonding interface temperature of 270°? is arrived at is 0.6 second, a bond strength of 5.1 pounds per inch results. If it takes, however, 2.4 seconds to obtain the same temperature, 270°F, a bond strength of only 0.5 pounds per inch is obtained. Although an explanation of this behaviour on the basis of some fundamental preperties of the material involved has not been found, a practical use of the indifference curves, which this writer be- lieves to be characteristic for each of the materials used in heat sealing, is possible. Since heat seal strength is only one aspect of a flexible package's performance, the other being its barrier properties, no attempt was made, at this stage, to devise a simulation system for design aptinatization. Until results of investigation of heat seal- ing conditions effect on the deterioration of flexible packaging mate- rials barrier properties vill be available, a procedure for quality control of heat sealing only, can be proposed. Utilisation of such a procedure presumes two factors. First, that the manufacturers supply together with other pertinent techs nicsl data, such as thermal conductivity, heat capacity. and pos- sibly the appropriate contact resistance, a set of indifference curves deve10ped for each of the materials they produce. Second, that the user, package fabricator, has access to, and ability to use a digital computer. Assuming that both the conditions are satisfied, the user adapts the Beck's computational model to conform to the physical 70 features of the heat sealing situation. by assigning appropriate values to the relevant parameters, such as thermal and physical con- stants of the film or films, the thickness of each, and similar con- stants for the components of the heatsealer in question. By selec- ting a feasible heated bar temperature T , be than generates a tem- perature function whose values enable the determination of the expected bond strength. It is obvious that such procedure could be used for a variety of purposes, from finding the proper temperature Th setting when the dwell time is invariable due to a fixed production speed of s machine and a desired level of bond strength is required. to determination of the dwell time necessary to achieve a given bond strength under fixed Th, or for purely predictive purposes. Care should be exercised not to be misled into regarding the indifference curves as the Th temperature which has no direct rela- tionship with the bond strength. If the simulation procedure were left entirely to a computer, the indifference curves would have to be stored in the computer memory which can be done by a system of ordered variable pairs (y,x) for each of the indifference curves, from which an intermediate value of y can be determined by means of any mnber of interpolating methods such as the Newton's Interpolation Hethod of Descending Differences. f(x+n) - (1+4 )“£(x) " {(3)413 df(x)Wf(x)W(fl+uo where n - fraction of dx A'flx) :- f(x+dx)~f(x) d‘flx) a &f(x+dx)-A'f(x) 71 - f(x+2dx)-2f(x+dx)+f(x) £“f(x) - A'f(n+dx)« A"f(x) - “n+3 dx) -3£(n+2dx)+3f(x+dx) -£(x) Another possibility is offered by representing each of the indifference curves by a power series which, because of the rec- ognizable hyperbolic characteristic, will take the form, A A A. A y-A +—L+ fi-‘i- .....+ 3" . o x i . ‘11 Although the derivation of coefficientsAn is rather lengthy, the final formulae offer the advantage of direct evaluation of the bond strength value y for any given 1 within the empirically defined interval. The expressions for the indifference curves of figure 21 are given below. 1 -4 o .. 03 1240 - 10.029 - 18.3293 + 13.112x 2 - 4.487: + .744: ‘ 00461.5 .250 i x ".500 I - 235.499 + 720.5528.1 - 925.100:.2'+ 647.910x.3 y250 . - 263.4%?“ + 56.311x‘5 + .556: 6 .325 i x ‘ .675 1 2 3 + 52438.81- - 19355.21? - 43064.9:‘5 + 15425.3:‘5 - 3110.0 - 19465.92” y260 + 73742.02?“ 0,000 ‘ X ‘ 1.200 1 -4 yzm - 7.596 - 51.2783" + 143.934x’2 - 202.3292'3 + 155.7541; - 61.160x'5 + 9.5642'6 .500 ‘ x ‘ 2.300 72 7275 - 42.0 - 533.51?1 + 3342.2:‘2 . 10120.73-3.+ 17507.0x-4 . 17243.6x‘5 + 8921.3x'6 - 1860.2x‘7 .650 ‘ x "3.450 ”230 - - .2564 + 4.6141x"1 - 7.4326x’2 + 13.4433x'3 " 5.7655164 .800 4 x i 3.200 1 3 .4 17.26 - 187.86x' +-789.25x'2 . 1510.52x' . 478 e 93‘.5 y285 - + 1376.90x 1.000 6 x ‘ 3.500 7290 - - 7.971 + 64.309x.1 - 133.7082?2 + 146.1742'3 . 62.843x"4 10200 ‘ x ‘ 3.200 2 3 “1 u a - ‘ 4.562 + 570554! " 96.5381: ‘2' 680275‘ y295 2.800 s x c 4.000 gecommegdations_§or ggture Study . The known relationship between the bulkiness of side groups of polymers having identical backbone molecular structures and the I“, as well as, the observation of the indirectly determined contact resistance which seems to follow a similar pattern, points out the possibility of existence of bulkiness-contact resist- ance relationship. As more metalized polymeric films become available, the resistance-molecular structure relationship should be studied with view on determining the contact resistance levels applicable for groups of films possessing similar structures. Further, if quantitative methods are to be employed in heat seal bond study and bond strength prediction, the investigation of mechanisms and rates of the diffusion process are necessary. As al- ready mentioned, radiography could be used in determining not only the ‘5! II II I... . 73 concentration and concentration gradient of the interdiffusing polymers but also in establishing the depth of penetration dur- ing bonding. The effects of pressure could be also studied by the above techniques. The results may have considerable consequences, such as lightweight construction and thus higher speeds of packag- ing machinery, if pressure is found to affect adversly the bonding rate or the depth of penetration of interdiffusing films. 1. 2. 3. 4. S. 6. 7. 8. 9. 10. 11. 12. 13. LIST 01" REFERENCES Scarpa, Thomas J., "Joining Plastics with Ultrasonics," Pgstic Technolggz, January 1962. Stuve, Diplnlng... Gerhard, "Schweissen und Reins-Siegeln von Kunststoff-Folien und Kombinationsfolien," Die Neue ESEEEEEEES’ 'ble 15 (1962),IN01 ‘0 Voyutskii, 8.3.. "Adhesion and Autohesim of Polymers ," dhes v Age, April 1962, p. 30, taken from Deryagin, B.V., and ILA. lrotove, Dokladz Akad, Nauk §SSR, Vol. 6 (1948). Ibid.. p. 30, taken from Skinner, 8.0., Savage, R.l.., and LE. Rutzer, Dokladz Akad, Nauk SSSR, Vol. 105 (1955). Ibid., p. 31, taken from Voyutskii, 8.8., Shapovalova, A.I., and 1.2. Pisarenko, Dokladx Akad, Nauk 8835, Vol. 105 (1955). Ibid., p. 31, taken fran Horosova, L. P., and LA. Krotova, 201. Ed y &d , Egg 8§SR, Vol. 115 (1957). Ibide. Po 35. Nielsen, Lawrence 3., geghanical Properties og Pglmrg, Reinhold Publishing Corporation” New York, 196 . Schricker, Gerhard, "Ueber die Bestimung der Pestigkeit von Reiss- Siegelnaehten," We, '01. 41 (1931), lo. 6. Ninnemen, IJL, "An Improved Laboratory neat Sealer," godeg Pac - 533.3,, November 1957. Kavesh, Sheldon, "Heat Sealers Need Insulation, Polypropylene study Show." W October 1964. Melvey, J.H., and I1.11. Stone, "Scalability of Polyethylene Films," Plastic Engineering, June 1959. Schneider, P.J.. Qanductiojntgeag gransfer, Addison Wesley Publish- ing Company. Inca. Reading.:M38°e. 1957c Akers,3rian I... "Surface Temperature Measurement in Thin Plastic Films," 3 es Michigan State University, School of Packaging, 1965. (unpublished) 7h 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 75 Anon., "For Heat-Sealing Strength and Characteristics ,” Packaging Institute Standard Test Method - 3, Ederg Packaging, September 1946. Vasenin, 3.11., "Adhesion of High Polymers," Adhesive Age, May 1965, pp. 18-25, June 1965, pp. 30-5. Crank, 1., me Mathematics of Diffusion, Oxford University Press, London 3.0. 4, 1964, p. 228, taken from Darken, L080. 3118 Am ”1111 He E IS NO. 175 (1948). Rabak, William, and 1.8. Stark, "Sealing Temperature and WP," Hodern Packaging, Vol. 19 (1946), No. 8. Rabat, William, and 0.1.. Dehority, "Effects of Heat Sealing on Water Vapor Permeabilities of Coated Cellophane," E. gdcrg PaCkagmg. '01s 17 (1944). Nos ’0 Jastrzebski, Zbigniew 0., Eggs and Prmrties cg Egmerigg Eaterials, John Wiley and Sons Inc., New York .- London, 1962. Daniels, P., and Robert A. Alberty, Physigal Chemigm, John Wiley and Sons Inc., New York - London, 1962. 7‘1 Severe, 8.2., Rheol of Pol , Reihold Publishing Carpets- tion, New York, 1962. Nielsen, 22, cit. Carslaw, 11.8., and J.C. Jaeger, Conduction of Heat in 59119;, Oxford University Press, London 2.0. 4, 1959. Rational Research Council of the U.S.A., Qternstiml Critical ables of Humerica Data h ics Chemistr and Technol Vol. VL, McGraw Hill Book Company Inc., New York, 1929, Po 1360 Griskey, 91.6., and 1!. Waldman, "Isotaetic Polypropylene - Thermodynamic Properties of Polymers «- Part 3," Modern W April 1966. APPENDIX 1 Models APPENDIX I Model. I Figure 22: Schematic Representation of Heat Sealing Assembly for Model Is A B C XII-0 and. A a- heated metal bar 3 - film 0 - mhaated metal bar Problem: .(E. a (fig 9 0 ‘ I ‘ L dt dx "t1.h’8tx.°.‘)o T-‘l‘z, ata-L,t>0 l'I-‘l'o-constantatt‘O Solution: Equation 1 2E2 a 2 2n “#5.. L2 +flZ-m9g-‘za’uf‘u [w] Q hoes a". 2L .. . a .2. rant) r1+ (1'2 21)L+n, 77 78 where T(x,t) - temperature at x and time t L - total thickness of film Th - temperature of heated bar H I initial film temperature 2: 0 thermal diffusivity of film ~ n - index of summation. 79 M06 Figure 23: Schematic Representation of Heat Sealing Assembly for Model 11. r1 r2 10-0 no A - heated metal bar 3 - film ( owl-infinite slab ). Problem: a: “dc 11> ’1‘2 T-‘l‘lnl'h. eta-O, t>0 T-‘l‘z-l'o, eta-o, tlo Solution: Equation II “rm 2 ”‘2 o 1 2 «s whersfi/ ' dz- erftz) - W o 1" 2 Th :- temperature of heated her To -- initial tanperature of the £11m 0‘ - thermal diffusivity of film. 80 do I Pigure 241 Schematic Representation of Heat Sealing Assembly for “Odel III. - Th ‘1' A . heated metal bar 3 - sir layer C - film ( semi-infinite slab ) . fl-lflI-O l “L‘x‘0,t70 d (11' d3: 26: (IT 61' K.’.'..la .J. at. -0 :70 ldx l2:11: x . x70,t>0 Tl-Tz atx-O, t>0 Th? To Solution: Equation 11! 2 «1* °° Izmt) - ‘1'o + (Th JZB“ erfc W 1+O‘ . 2 0‘1: where 12(x,t) - temperature within the film ‘1'h - heated bar temperature. 81 1'0 - initial temperature [1,K2 - thermal conductivities of air and film «1.0!: - thermal diffusivities NF 1" r6?" H 9" 0-- ~- . K 1%.. (a... q +1 erfc - l . erf 82 Mall! Figure 25: Sebastic lepreseetstion of last Sealing Assembly ‘0: m1 “0 . A s c s i s r c x ai b c 966*7 ffiLv .___.‘ A I- unheated metal bar I 1' insistiesvnsteriel 3 . fil- D - air layer I - fence eeetisc P - met-semis 0 - heated metal bar a,b,e,d,e,f,g - thicknesses of the mu. Problem the problem still is to solve the ted sendeetia equation I if: . .1 n .. o , u a: 832 the approach used, hon-ever, is the Crank-lioholson hthod of linen finite Differences Apprailstiu fiieh uploys the "heat balance”, i.e., heat increase in a section - heat in - beat out. the solutial is obtained from a system of simlteneous difference equation given below. 2 e ‘59 1:: III, - i . .0 ‘I .I 3. is . ~ ole, ‘ _ .. 83 Figure 26: Nodal Scheme for Internal Nodes. n:l n M 1 O . p—————-AI——-—~—e~ Equatim 4‘8 e ‘ In I 01‘ 1 £35 ”Ea-n ea '24—? K n I .1 --;- g; rn-i‘ul «21;: J2“ 3%112; .clfld Figure 27: Nodal Scheme for Interface Nodes. i 2 3 4 ' A x 1| :1 -——-—d-s--—h —dl—--—-—- ‘2 ~———-——-—e-1 Eqmtmn ‘.be Til. i‘l "----;—-L:-——1-—-)~ PC (Mt )2’-1"21lx +2hAx1+__L_1_L.—___)~ ’00 453,921 + 1'; ZhAxl m-l ’z’:°z‘“1)2l . 1"1-’l'°1c1“°"1’2 I " 1 1" A‘ "’ 2 for material with K1. . rib...“ 84 where ‘l' . temperature At - time increment Ax :- dietance between nodes node index time index In Pi - density of ith materiel 01 - heat capacity of ith material K1 :- thermal conductivity of ith material h - contact resistance (hi3 . l. - thickness of air layer )1 - boundary approximation ometant APPENDIX II TABLES APPENDIX 1! TABLE 1 Computed Aluminum Resistivity as a Function of Temperature Temgereturg Deg. a? Resistivigz g 106 ohm.- gm 100 3.068 110 3.135 120 3.202 130 3.270 140 3.338 150 3.406 160 3.é75 170 ‘ 3.543 180 3.613 190 _ 0 3.682 200 ' 3.752 210 3.822 220 I 3.892 230 3.962 240 6.033 250 ' ‘ ' 0.106 86 87 IABLE 2 Thermistor Temperature Calibration Date for'Vsrying ‘Loeding Levels at Sensitivity Setting of .02 Volts per Division Temperature Loading W13 1&1170 [SQe ins Me 03 W 022 e99 1e77 2e54 3.32 100 *1.32 1.36 1.42 105 1.52 1.46 110 1.79 1.90 1.88 115 2.19 2.00 120 2.32 2.43 2.39 125 2.68 2.47 130 2.78 2.87 2.70 135 3.09 2.99 140 3.34 3.41 3.25 145 3.52 3.49 150 3.80 3.86 3.95 155 4.00 4.10 160 4.36 4.22 4.30 165 4.50 4.60 170 4.80 4.71 4.89 175 5.15 5.20 180 5.41 5.22 5.50 185 5.67 5.58 190 5.82 5.70 5.93 195 6.18 6.06 200 6.20 6.30 6.38 205 6.45 6.60 210 6.84 6.88 6.74 215 7.19 7.21 220 7.39 7.29 7.42 225 7.71 7.60 230 7.78 7.89 7.92 235 8.00 8.15 240 8.36 8.45 8.30 245 8.56 8.70 250 8.82 8.83 8.98 e Tabulated entries represent divisions on oscilloscope screen. 88 TABLE 3 Temperatures Determined Expertmentally Thermistor Tickness - .001 inch 1' - 80°F. 1' - 250°F o h Tenperature at .001 inch away Time flee from Heated Bar 0.00 80.0 0.05 170.5 0.10 182.5 0.30 206.5 0.50 214.0 0.75 219.2 1.00 222.1 .1.50 225.5 2.00 228.5 TABLE 4 Temperatures Computed by Equation 1 To ' 80°F ’ Th . 250°? time See temperature at x - .001 inch Deg. 9? 0.000 80.0 0.002 115.2 0.004 142.1 0.006 154.5 0.008 160.2 0.010 162.8 0.012 163.9 0.014 164.5 0.018 _ 164.9 0.020 165.0 on 165.0 In. 1" in. .117 . 89 TABLE 5 Temperatures cmlputed by Equation 2 o rec-30°17, rh-zsor Temperature at x u .001 inch ‘rins lee Dec. 9! 0.00 80.0 0.02 197.3 0.04 212.3 0.06 219.1 0.08 223.1 0.10 225.9 0.20 232.9 0.60 240.1 1.00 242.4 1.40 243.5 1.80 244.3 2.20 244.9 .v. TABLE 6 Temperatures Computed by Equatial 3 '1'0 - 30°11 , rh - 250%. 1. - .0001 inch Temperature at x - .001 inch Time Sec Deg. 0? 0.00 80.0 0.02 164.6 0.04 183.7 0.06 192.8 0.08 198.4 0.10 202.2 0.30 216.4 0.60 222.2 1.00 225.3 1.50 227.3 2.00 228.5 90 TABLE 7 temperature Coeputed by Equation 4 with Finite Thickness of Insulation '1'o - 80°F, rh - 250°r, 1. - .0001 inch Temperature at x - .001 inch Time Sec Deg. 0.00 80.0 0.02 144.5 0.06 170.6 0.10 187.1 0.20 207.8 0.40 220.4 0.60 224.7 0.80 227.3 1.00 229.1 1.50 231.9 2.00 233.6 TABLE 8 Temperature Computed by Equation.4 with Finite Thickness 0! Insulation and .001 inch‘reflan Coating o ro-so°r.rh-2sor . Time ‘renpereture at x - .001 inch in deg.2l 0.00 80.0 80.0 80.0 0.10~ 179.8 164.9 144.7 0.20 197.1 184.4 164.8 0.30 204.9 193.8 175.4 0.50 213.3 203.9 187.6 0.70 217.9 209.7 194.9 0.90 221.1 213.6 199.9 1.00 222.3 215.1 201.9 1.50 226.4 220.4 209.1 2.00 228.9 223.6 213.5 91 3.94 1.. vv-w—rw—w— W L n.‘ _._.4... - - v," 4..“ 1 'Entries in lb/in are averages of ten samples TABLE 9 Bond Strength in Pound per Inch Tabulated for Each Heat Sealing tins and Pressure Combination and Varying Heated Bar‘Temperature Th. Pressure Heat Sealing Time in Sec neg.°r P"'1‘ .2 .5: 1.0771;671 2.0 3.0 4.0 300 1.68 .07 .10 .11 .19 .24 .23 .26 1.15 .12 .13 .15 .21 .20 .21 .24 1.61 .18 .37 .39 .49 .47 .52 .49 2.09 .14 .22 .29 .36 .35 .34 .34 2.55 .13 _.13 .22 .28 .31 .33 .32 3.02 .09 .19 .23 .26 .29 .28 .30 3.47 .09 .14 ..19 .23 .26 .29 .27 3.94 ‘ .08 .13 .21 .23 .27 .29 .28 315 p .68 .27 .64 .86 .98 .92 .88 .91 1.15 .54 .96 .94 .99 1.08 .98 1.11 1.61 .59 .84 1.18 1.51 1.56 1.48 1.51 2.09 .51 .78 .98 1.21 1.32 1.36 1.37 2.55 .41 .80 .98 1.18 1.43 1.46 1.43 3.02 .40 .63 .97 1.22 1.38 1.22 1.37 3.47 .39 .57 .92 1.06 1.36 1.28 1.41 .37 “.57 '.76 .98 1.26 1.39 1.32 TABLE 9 _ (Continued) Th Pressure o 8:2— o‘ad‘ 821‘ 111-2 __g__)_.,_d3_§.o; x>0,t>0 dxz “I ‘15 Observing that 1" - 10+ Va 1" - To-I- vf '1' - 'r + V o 97 98 where To n initial temperature of the system vi - variable temperature in air layer v: I variable temperature in film material 1' - constant temperature of the heated bar at x a-l V - constant temperature gradient between I --1 and ‘81- or. We can not: treat the whole problem as a solid with zero initial tem- perature and x I 01 kept at constant temperature‘V for t >-0. the differential equations to be solved become: 2 111-18 dvg__k:'_1-oi ~14x<0.t>0 dxz ex. dt 42v dv “1‘2“ J—iJ-O; x>0,t70 d‘z C5£4it Boundary conditions are: dv dv III-3 Rafi-K131; x-0.t>0 III-4 v. - v: z x . O . t:> 0 Applying Laplace transformatim to III-1a and III-23 yields the subsidiary equations: III-5 d'3_qz§ -o; ~1 () 1 m a we get 2 °° 2 1 . I, tri- + bx . III 18 vi fizfln erfc 2 o( t o a Similarly w 111.19 Va I V Z/Sn [aria 2 + 1 + o (Serf: 2 + . It]. a 2 an: 2 Ida: Since we are interested in temperatures within the film material, we substitute the relations I . 1'0 + vf, and r - To + V 2 into 111-13. to obtain the desired solution: 2cm) °° III-20 T£(x,t) .. 0 fin erfc W] + To . 1 + a- 2 [T t The solution is now complete. "7'1! @1117 4791444 if '14 FF