Appendix A Measures of Strain for Large Deformations
A.1 THE DISPLACEMENT FUNCTIONS AND THE DISPLACEMENT GRADIENT TENSOR As the first step in establishing a quantitative measure of the strain that occurs in a fluid as a result of a large deformation, we need to describe the position of a particle of fluid as it moves about during a deformation process. This is easily done by writing down the coordinates of the particle as functions of time. However, we need to keep track of which particle we are tracking. This can be done by "labelling" each particle with its position vector, x, at some refer• ence time, t 1• Thus, giving the components of x(t1) identifies a particular fluid particle and distinguishes it from all other particles. At some other time, t 2 , the coordinates of this particle will be given by the following "displacement functions":
Xl = Xl [ t 2' x{t 1)] (A-Ia) X2 =X2[t2,X(t1)] (A-Ib) X3 = X3[t 2,X{t1)] (A-Ic)
These functions tell us the location at any time t2 , of the particle that is located at X(t1) at time t 1• As an example of the use of the displacement functions to describe a deformation, consider simple shear flow in which the shear strain, 1', is some specified function of time, y(t), with 1'(0) = O. The displacement of a typical fluid particle is shown in Figure A-I. We locate the origin for the position vectors of all fluid particles at point 0 on the lower, stationary plate. Since we shall be considering only one or two specific particles in the remainder of
601 602 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
X,(t2)-j X, (t'):::;:::i ! T X2(t,)
o
Figure A-I. Fluid particle displacement in simple shear.
our discussion, it will not be necessary to include the particle label, X(t1)' as an argument of the displacement functions. Thus, the shear strain can be expressed in terms of the displacement functions as follows for the particular particle shown:
x j (t2) - XI(t l ) (A-2) y(t2) - y(tl) = X2(t l )
where [X I(t2) - xj(tl )] is the distance moved in the XI direction by a fluid particle that is at a distance x 2 from the stationary plate, during the time interval between t1 and t 2. The displacement functions for this deformation can then be expressed in terms of the shear strain as follows:
X j (t2) = x 1(t j ) + X 2(t 1)[ y(t2) - y(t1)] (A-3a)
X 2(t2) = X 2(t l ) (A-3b)
X 3(t 2 ) = X 3(t l ) (A-3c) To describe the deformation of a fluid element, we need to examine the relative displacement of two fluid particles. Consider two neighboring particles that are separated by the vector dx(t I) at time tl and by dX(t2) at time t2, as shown in Figure A-2. If we could define a quantity relating these two vectors, it might be useful to describe the strain that has occurred between times tl and t 2 • One such a quantity is the "displacement gradient tensor," F;/tl' t2), which is defined as follows:
3x;{t2 ) Fij(t p t 2 ) == () (A-4) 3xj tl APPENDIX A 603
t--I,-----IlX,------l
,------, t,
o
Figure A-2. Position and displacement vectors for two particles of an element of a body of fluid undergoing solid body translation.
Another tensor that relates the two vectors dx(t2 ) and dx(tl) is the "inverse" of the displacement gradient tensor, which is defined as follows l :
(A-5)
The use of the displacement gradient tensor to determine dx(t2 ), given dx(tl), is demonstrated below:
dx j (t2 ) = Fil(tl , t2 ) dxl(tl ) + Fi2(t!> t2 ) dx 2(tl) + Fi3(tl , t2 ) dx 3(tl ) (A-6)
The inverse tensor can be used in a similar manner to determine dx(tl) given dx(t2 ). lNote that the inverse of a tensor is not the tensor whose components are the reciprocals of the components of the original tensor. 604 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
To see if these tensors are useful as measures of strain, we consider first the "solid body translation" of the fluid element shown in Figure A-2. In this type of motion, there is no deforma• tion of the fluid element, i.e., no strain. To see if the displacement gradient tensor or its inverse reflect this, we first write the displace• ment functions for this motion:
XI(t2) = Xl(t l) + dX1 (A-7a)
X2(t2) = X2(t1) - dX2 (A-7b)
X 3 (t 2 ) = X 3(t 2 ) (A-7c)
Now we want to evaluate Fij and its inverse for this flow. We note that the two vectors dx(t2 ) and dX(tI) are equal; this implies (from Equations A-4, A-5, and A-7) that the components of F;/t l , t2 ) and its inverse have the special, simple form shown below: o 1 (A-8) o ~l where the quantity on the right is called the "unit tensor." When a vector is multiplied by the unit tensor it is unchanged, and since
t = t,
o·~------x' o~------x,
Figure A-3. Displacement vectors for two particles of an element of a body of fluid undergoing solid body rotation. APPENDIX A 605
there is no strain in this example, that is exactly the way we want a strain tensor to behave. Thus, both FJtl , t2) and its inverse show promise as possible measures of finite strain. As a further test of the usefulness of these tensors as measures of strain, consider the "solid body rotation" shown in Figure A-3. Again there is no deformation, i.e., no strain. However, in this case the vectors dX(t2) and dx(tl ) are not equal and neither Fi/t l , t2) nor its inverse reduces to the unit tensor. Thus, neither of these tensors is, after all, a useful measure of finite strain.
A.2 THE CAUCHY TENSOR AND THE FINGER TENSOR There are other tensors, however, that do have the desired proper• ties and are thus useful as measures of strain for large deforma• tions. Two of these are the Cauchy and Finger tensors, which are defined as follows: (A-9)
(A-10)
A.3 THE DISPLACEMENT GRADIENT TENSOR FOR SIMPLE SHEAR AND SIMPLE EXTENSION
In order to derive the components of the Cauchy and Finger tensors for simple shear and simple extension (Equations 3-1 to 3-4) it is useful to have available the components of Fij and its inverse for these two deformations. For simple shear flow the components of these tensors are: ~l (A-H) ~l (A-12) The components of the Cauchy tensor for simple shear (Equation 3-1) are obtained by combining Equation A-9 with Equation A-H, 606 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
t F
Figure A-4. Principal component of the position vector for simple extension of a sample with one end fixed at Xl = O. while the components of the Finger tensor (Equation 3-2) are obtained by combining Equation A-lO with Equation A-12. For simple extension, with x I taken to be the principal stretch direction (see Figure A-4), the distances of a fluid element from the Xl = 0 plane at times tl and t2 are related to the Hencky strain, e, evaluated at these two times, by:
(A-13)
The displacement functions are:
X1(t2) = xl(tl)exp[ e(t2) - e(tl )] (A-14a)
X2(t2) = x 2(t1)exp{( -1/2)[ e(t2) - e(tl)]} (A-14b)
X3(t2) = x3(tl)exp{( -1/2)[ e(t2) - e(tl )]} (A-14c)
And the components of the displacement gradient tensor and its inverse are: o e{( -1/2)[E(t2)-E(tj)]} o (A-IS)
(A-16) Appendix B Molecular Weight Distribution and Determination of Molecular Weight Averages
Synthetic polymers are not composed of chains of equal lengths. Rather, they have a distribution of lengths or molecular weights. The exact form of the distribution depends upon the details of the polymerization mechanisms, and may also be affected by subse• quent treatment. Treatment with solvents can selectively remove low molecular weight fractions. Shear induced degradation can preferentially cleave the higher molecular weight components. Chemical reactions such as oxidation can cause either chain scission or crosslinking. Fractionation according to molecular weight can occur during complete precipitation from a solution, so that the average molecular weight of a polymer in powder form may depend on the particle size distribution of the powder. Various physical properties, including rheological properties, have different dependencies on molecular weight distribution (MWD), and it is therefore important to be able to describe and measure the MWD. Conversely, the determination of the MWD curve is done by measuring properties that have a known dependence on molecular weight. The most commonly used techniques include light scatter• ing from dilute solution, dilute solution viscosity, dilute solution osmotic pressure, and rate of elution from a porous medium (gel permeation or size exclusion chromatography-GPC or SEC) [1,2]. A distribution can be described by specifying the weight fraction of the polymer that has a given molecular weight. (It can also be specified by the number fraction of a given molecular weight; the two specifications are easily interconverted). Such a specification
607 608 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
5.0
4.0 '"o,.... X z B3.0 ~ II: L1. I- a 2.0 iii ~
1.0
o.o~~ __.. ~~~~~~~ .. ~~~~~~ __~~~ 10 100 1000 10000 1 ooooq 1 000000 MOLECULAR WEIGHT
Figure B-l. Differential molecular weight distribution curves for most probable (solid curve) and log normal (dashed curve) molecular weight distributions, both with Mw/Mn = 2.
can be presented in the form of a diagram like that shown in Figure B-l. It is usually more convenient to represent the MWD by a smooth curve that can be represented by a mathematical function. Often such a function can be derived from a knowledge of the kinetics of the polymerization. It is most convenient if this function contains only 2 or 3 parameters, which can then be expressed in terms of the commonly measured average molecular weights. For any distribution function one can define various averages. For example, if the polymer contains a weight fraction wi(M) of polymer of molecular weight M j , the weight average molecular weight M w is defined as
IMw.(M.) M = I I I (B-1) w Iwj(MJ APPENDIX B 609
By definition of the weight fraction, the denominator of Equation B-1 is unity. The summation extends over all possible values of the index i. It should be noted that the distribution wi(M) is often expressed in the literature as a continuous function w(M) dM, the weight fraction of molecules of molecular weight between M and M + dM. In that case, and similarly for the other averages dis• cussed below, the summation is replaced by an integral. To illus• trate, for Mw this takes the form
00 i Mw(M)dM M w = --"-0-00----- (B-1a) w(M)dM io
The number average molecular weight Mn is defined similarly by a summation of Mi over the number fraction ni(MJ Since ni(M) is related to wi(M) by the expression ni(M) = wi(Mi)/Mi, this average can be expressed in terms of the weight fraction distribu• tion as (B-2)
Various other higher molecular weight averages can be defined similarly. Two that are important in rheology are:
(B-3)
(B-4)
In order to calculate the average molecular weight of a blend of polymers, for each of which the averages are known, it is necessary to sum the weighted numerators and denominators of these expres• sions individually [3]. This leads to the following blending rules; the symbol As shown diagramatically in Figure B-1, the different averages are weighted more or less heavily by the low and high ends of the MWD, the Mn being smallest and Mz+ 1 highest. As mentioned above, the mathematical form of the MWD de• pends on the chemistry of the polymerization and any subsequent treatment. The reader is referred to standard polymer chemistry textbooks, including the classic one of Flory [4] for details. Two of the most important MWDs are given here. The "most probable" MWD has the form: (B-5) The successive average molecular weights for this MWD are in the ratios The most probable MWD is the one that results from polyconden• sat ion reactions, such as those for Nylon-6,6 and polyethylene terephthalate, and for other polymerizations involving a randomiza• tion step, as for acetal copolymer. A more complicated MWD is the "log-normal" distribution, so-called because the plot of w(M) against the logarithm of the M APPENDIX B 611 follows the Gaussian normal error curve. It has the form: (B-6) Mo characterizes the location of the distribution and /3 its breadth. The weight average molecular weight M w is and the ratios of molecular weights are This MWD is encountered with polymers such as HDPE and polypropylene that are produced by polymerization with a hetero• geneous catalyst. Depending of course on the value of /3, the MWD can be extremely broad. A typical HDPE can have, for instance, an Mn of 10 4, an Mw of 10 5, an Mz of 10 6 , and at least theoretically, an Mz+ 1 of 10 7• As for methods of determining the MWD, the reader is again referred to polymer textbooks. However, briefly, Mn is measured by methods that count the number of molecules, such as osmotic pressure of dilute solutions or chemical or spectroscopic end group determinations. Mw is measured by light scattering, which can also give an indication of M z • Undoubtedly the method of choice for determining the complete MWD is GPC, which does, however, require calibration, usually with narrow MWD polymers whose molecular weights have been determined by one of the above absolute techniques. A very convenient measure of molecular weight is the dilute solution viscosity. The quantities usually used to describe the solu• tion behavior are the intrinsic and inherent viscosities, which are defined in Appendix C. Again, this requires calibration, and if the polymer of interest has a very different MWD than the calibration material, the molecular weight average determined will be some• what in error. Generally, however, it will be close to Mw. 612 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING One other caution is that long chain branching affects both the GPC elution volume and the intrinsic viscosity. Comparison of a variety of measurements is therefore required to establish the existence of and to estimate the amount of long chain branching. This is by no means a trivial exercise, and should probably be considered to be more of a research project than a routine measurement. REFERENCES 1. A. R. Cooper, Polym. Eng. Sci., 29:2 (1989). 2. S. Balke, Quantitative Column Liquid Chromatography, Elsevier, New York, 1984. 3. K. F. Wissbrun, Trans. Soc. Rheol. 21:149 (1977). 4. P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953. Appendix C The Intrinsic Viscosity and the Inherent Viscosity C.1 DEFINITIONS OF SOLUTION VISCOSITY QUANTITIES The most common method for determining an approximate average molecular weight for a polymer is to measure the viscosity of a dilute solution [1]. The measurement is usually made using a glass capillary viscometer.! Several quantities are used to describe the viscosity of a solution, and we define these below. When two names are given, the first is that proposed by Cragg [2], and the second is that proposed by IUPAC [3]. The relative viscosity or viscosity ratio: (C-l) where 'TIs is the solvent viscosity. The specific viscosity: 'TIsp == 'TI r - 1 (C-2) The reduced viscosity or the viscosity number: (C-3) where C is the concentration in g/dl (g/lOO mI). lA vibrational viscometer such as that made by Nametre can, in principle also be used. It is faster and more convenient but requires a larger sample and is somewhat less precise. It is also more difficult to control the temperature of the sample. 613 614 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING o CONCENTRATION, gjdl Figure C-l. Sketch showing the general behavior of two functions of the solution viscosity as functions of concentration. Both converge to the intrinsic viscosity in the limit of zero concentration. The inherent viscosity or the logarithmic viscosity number: (C-4) This quantity is not to be confused with the "inherent melt viscos• ity" defined in ASTM D3835 [6]. The latter is the result of an extrapolation technique designed to eliminate the effects of thermal degradation in the use of a capillary rheometer to determine melt viscosity. The intrinsic viscosity or the limiting viscosity number: (C-5) While both the reduced and inherent viscosities approach the intrinsic viscosity in the limit of vanishing concentration, they di• verge at larger concentrations, as indicated in Figure C-l. For values of [ '17] of about 2 or higher, the solution may be shear thinning, and in this case an extrapolation to zero shear rate is required to obtain the intrinsic viscosity [4]. For ultrahigh molecular weight polyethylene, a special rotational rheometer has been rec• ommended to obtain data for such an extrapolation [5]. APPENDIX C 615 C.2 RELATIONSHIP TO MOLECULAR WEIGHT The use of solution viscosity to provide information about molecu• lar weight arises from the observation that for linear, mono disperse polymers, the intrinsic viscosity is related to the molecular weight as follows: (C-6) where K' and a depend on the polymer, the solvent and the temperature. Extensive tabulations of the constants have been published [7-9]. Equation C-6 is called the Mark-Houwink-Sakurada equation. In the case of polydisperse materials, one can use this relation• ship to define a "viscosity average molecular weight": (C-7) It can be shown that this average is related to the molecular weight distribution as follows: (C-8) When a = 1, Mv becomes equal to Mw' Often Mv is about 80 to 90% of Mw , and an often-used approximation of (C-7) is: [71] = k'M~ (C-9) In fact, k' and a in this equation are not truly independent of molecular weight, and Manaresi et al. [10] have proposed a correc• tion term for (C-9) that depends on Mn, Mw and Mz. C.3 RAPID TESTING FOR QUALITY CONTROL PURPOSES We note that the determination of the intrinsic viscosity requires measurements of the viscosities of several solutions and an extrapo• lation to zero concentration. For routine quality control, a quicker and more common practice is to simply measure the inherent viscosity (IV.) at some standard value of the concentration, usually 616 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING between 0.1 and 1 gldl and often around 0.5. While this does not lead directly to a quantitative determination of the average molecu• lar weight, it does provide quick evidence of a variation in Mw from one sample to another. REFERENCES 1. F. W. Billmeyer, Jr., Textbook of Polymer Science, Second Edition, John Wiley & Sons, N.Y. 1971. 2. L. H. Cragg, J. Colloid Sci. 1:261 (1946). 3. International Union of Pure and Applied Chemistry, J. Polym. Sci. 8:255 (1952). 4. B. H. Zimm and D. M. Crothers, Proc. Nat. Acad. Sci. 48:905 (1962). 5. H. L. Wagner and J. G. Dillon, J. Appl. Polym. Sci. 36:567 (1988). 6. American Society for Testing and Materials, "Standard Test Method for Rheological Properties of Thermoplastics with a Capillary Rheometer," D3835-79 (Reapproved 1983), Appendix. 7. M. Kurata and W. H. Stockmayer, Adv. Polym. Sci. 3:196 (1963). 8. M. Kurata, Y. Tsunashima, M. Iwama and K. Kamada, Chapter IV, p. 1, Polymer Handbook, Ed. by J. Brandrup and E. H. Immergut, Second Edition, John Wiley & Sons, N.Y. 1975. 9. N. C. Billingham, Molar Mass Measurements in Polymer Science, John Wiley & Sons, N.Y. 1977. 10. P. Manaresi, A. Munari, F. Pilati and E. Marianucci, Eur. Polym. Joum. 24:575 (1988). Appendix D The Glass Transition Temperature The temperature and pressure dependencies of the viscosity are closely related to a characteristic temperature, Tg • For the sake of simplicity, we consider a polymer such as ordinary (atactic) polystyrene whose structure is not sufficiently regular to permit packing into a crystalline form. At a very low temperature it is a rigid solid. If we heat this solid and measure its specific volume (the reciprocal of the density), we find that it increases, as does that of all solids, because of the increased thermal motion of its atoms. The thermal expansion coefficient a is defined as a = ~ ( dv ) (D-l) v dT where v is the specific volume and T the absolute temperature. Typically a is on the order of 2 X 10 - 4 K - 1. As we continue to heat the material we find that in a very narrow temperature range the thermal expansion coefficient begins to increase rapidly and becomes two- to three-times greater. This is illustrated in Figure D-l. The intersection of the straight lines that approximate the specific volume above and below this temperature range is called the "glass transition temperature" (Tg). Below Tg the material is a hard, glassy solid; above Tg it softens visibly and becomes a viscous liquid. The Tg is not a sharply defined temperature. Its value depends on the rate of heating (or cooling) and on the previous thermal history of the sample. However, these variations are relatively small (on the order of 3°C for a ten-fold change of rate) and we can consider Tg to be a characteristic of the material. Other thermody• namic quantities, including the specific heat Cp , show a similar 617 618 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING UJ ::i! :::> o...J > () u::: U UJ Q. (f) TEMPERATURE Figure D-l. Glass transition temperature Tg , as inferred from the intersection of extrapola• tions of the two straight-line segments of the specific volume versus temperature curve. transItIon. Probably the most convenient way to measure Tg is by determining the change of Cp by differential scanning calorimetry (DSC) or differential thermal analysis (DTA). A number of theories have been advanced to account for the phenomenon of a glass transition. These are not described here; the interested reader may consult the literature [1]. Briefly, however, we can describe what is happening as follows. The volume of the atoms comprising the material is essentially independent of temper• ature. The observed increase of the specific volume as the polymer is heated is ascribed to the introduction of "free volume," vf' When the free volume reaches a critical value (on the order of 2.5% of the total volume), the chain segments of the polymer have sufficient room to move freely on the time scale of the heating experiment. As a result they are able to respond to a further temperature increase by increasing vf more rapidly. The ability of the segments to move freely is reflected in the softening that is observed above APPENDIX D 619 Tg • This motion is observable not only by the mechanical response, but also by other measurements of mobility, such as dielectric loss and nuclear magnetic resonance (NMR). Polymers that have a regular chain structure, such as linear polyethylene, polypropylene, and polyethylene terephtbalate also show the glass transition phenomenon. The observation is compli• cated, however, by their crystallinity, which preserves the solid-like character of the polymer above Tg • Polymers such as PET crystallize slowly and can be quenched into the amorphous glassy state by rapid cooling; conventional volume or specific heat measurements are then possible. Also, if the maximum degree of crystallinity is not too high, the Tg of the "amorphous fraction" of the polymer can be observed conventionally. For highly crystalline polymers such as PE or polyacetal, the Tg is more readily determined by methods sensi• tive to mobility, such as dynamic mechanical analysis. Since many polymers show transitions other than the Tg , these methods can lead to some ambiguity. The value of Tg depends strongly on molecular structure. It is increased by substituent groups that stiffen the chain, as is to be expected from the effect of chain mobility. Polarity, which increases interchain cohesion, also raises the Tg • A few representative exam• ples to illustrate these points are shown in Table 10-3. Extensive tables are available in the Polymer Handbook [2]. Van Krevelen and Hoftyzer [3, p. 99] also tabulate data as well as "group additivity contributions" from which the Tg of an untabulated polymer can be estimated. The Tg values listed in Table D-l are for high molecular weight polymers. The extra free volume associated with chain ends de• creases the Tg of low molecular weight polymers according to (D-2) For polystyrene K is about 10 5 [4, 5], so that the Tg of a polystyrene with an Mn of 10 4 is 10°C lower than that of a very high molecular weight polymer. Plasticizers also have a strong effect on Tg for the same reason, i.e. they contribute free volume. It should be noted that small amounts of solvent may be retained tenaciously by polymers and can cause appreciable lowering of the observed Tg • 620 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING Table D-1. Variation of Tg with Molecular Structure POLYMER STRUCTURE TiOC) Polyethylene -CHz-CHz- -125 Polypropylene -CH -CH- -13 z I CH 3 Polystyrene -CHz-CH• + 100 I cf> Polymethyl Acrylate -CHz-CH- +10 I COOCH3 CH3 I Polymethyl Methacrylate -CH -C- +105 z I COOCH 3 The Tg of a random copolymer or of a miscible blend of two polymers can be estimated by various mixture rules, of which the simplest is merely the weight or volume averaging of the component Tg values. Another expression averages the reciprocals, (D-3) On the other hand, block or graft copolymers or immiscible blends will exhibit the Tg's of both components, perhaps modified by finite block size or by partial miscibility. Considering the relationship of free volume vf to Tg , it is not surprising that Tg depends on pressure. The magnitude of this effect is to increase Tg about 1.5°C per 1000 psi (0.025°C per 10 6 dynes/cm2). This is about the value one would calculate from the compressibility of a polymer melt. Finally, to close this discussion of transition temperatures, we should point out that although the Tg is the lowest temperature at which we can consider a material to be liquid, in many cases polymers solidify at a higher temperature, the melting temperature Tm' Polymers vary widely in their ability to crystallize, in the maximum degree of crystallinity attainable from the melt, and in the sensitivity of their rate of crystallization to temperature and cooling rate. A rule of thumb applicable to undiluted homopoly• mers is that Tg is between 0.5 and 0.67 times Tm , where both temperatures are expressed in absolute (Kelvin) units. However, APPENDIX D 621 small variations of structure that affect chain regularity, and thus the ability to pack, can change Tm by large amounts. It is worth noting that Tm is very sensitive to small quantities of comonomer but relatively insensitive to diluents, whereas the converse is true for Tg • REFERENCES 1. A. Eisenberg, "The Glassy State and the Glass Transition," Chapter 2 in Physical Properties of Polymers, 1. E. Mark, ed., American Chemical Society, Washington, D.C., 1984. 2. W. A. Lee and R. A. Rutherford in Polymer Handbook, Second Edition, 1. Brandrup and E. H. Immergut, Eds., Wiley & Sons, New York, 1975. 3. D. W. Van Krevelen and P. 1. Hoftyzer, Properties of Polymers, Second Edition, Elsevier, New York, 1976, p. 99. 4. T. G. Fox and P. 1. Flory, J. Appl. Phys. 21:581 (1950). 5. T. G. Fox and P. 1. Flory, J. Polym. Sci. 14:315 (1954). Appendix E Manufacturers of Melt Rheometers and Related Equipment This list was compiled for the use of readers who wish to obtain further information about commercial melt rheometers. It was the intention of the authors to include all companies offering such equipment, and any omission is the result of oversight and not of any selection process. Inclusion in this list does not imply any endorsement or recommendation by the authors. The information given was verified in 1989 but will naturally become less accurate with the passage of time. Bohlin Reologi Inc. 2540 Route 130, Suite 105 Cranbury, NJ 08512 (609) 655-4447 VOR Rotational Rheometer (cone-plate with torque and normal force measurement) C. W. Brabender Instruments, Inc. P. O. Box 2127 South Hackensack, NJ 07606 (201) 343-8425 Plasti-Corder (torque rheometer) Extrusiograph (laboratory extruder) Carri-Med Ltd. Glebelands Centre, Vincent Lane Dorking, Surrey RH4 3YX England 622 APPENDIX E 623 North American Representative: Mitech Corporation 1780 Enterprise Parkway Twinsberg, OH 44087 (216) 425-1634 CSL Rheometer (constant stress rheometer for creep and recovery) Weissenberg Rheogoniometer (cone-plate with torque and normal force measurement) Carter Baker Enterprises ltd. P. O. Box 2 Stansted, Essex CM24 8JG England (0279) 814810 ACER Series 2000 Capillary Extrusion Rheometer Ceast U.S.A. Inc. P. O. Box 3072 Fort Mill, SC 29715 (803) 548-6093 Rheovis 2100 (capillary rheometer) Modular Flow Index (melt indexer) Costech Associates, Inc. 1184 Corner Ketch Road Newark, DE 19711 (302) 239-2207 Costech 2000 Process Simulator (biaxial inflation rheometer) Custom Scientific Instruments, Inc. 13 Wing Drive Cedar Knolls, NJ 07927 (201) 538-8500 Melt Index Apparatus CS127 Melt Elasticity Indexer Model CS245 (concentric cylinder recoil apparatus) 624 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING Daventest Limited Tewin Road Welwyn Garden City Herts AL7 lAQ England Davenport Melt Viscosity System (gas-driven capillary viscometer) Davenport Melt Flow Indexers Dupont Company Instrument Systems Concord Plaza, Quillen Building Wilmington, DE 19898 (302) 772-5500 983 Dynamic Mechanical Analysis System 943 Thermomechanical Analyzer 2970 Dielectric Analyzer Gottfert Werkstoff-PrOfmaschinen GmbH Postfach 1220 6967 Buchen Federal Republic of Germany North American representative: Goettfert Incorporated 488 Lakeshore Parkway Rock Hill, SC 29730 (803) 324-3883 Viscotester 1500 (bench-top capillary rheometer) Rheograph 2002 (high-pressure capillary rheometer) Melt Indexer Bypass-Rheograph (on-line capillary rheometer) Side-Stream Rheograph (on-line capillary rheometer) Rheostrain (extensional rheometer) Karl Frank GmbH Postfach 1320 D-6940 Weinheim Federal Republic of Germany APPENDIX E 625 North American representative: Carl G. Brimmerkamp & Company, Inc. 102 Hamilton Avenue Stamford, CT 06902 (203) 325-4101 Melt Indexer High-Pressure Capillary Rheometer HBI/Haake Fisons Instruments 244 Saddle River Road Saddle River, NJ 07662-6001 (800) 631-1369 (In NJ: 201-843-2320) Rheodrive System System 90 (torque rheometers and extruder-fed capillary rheometers) Instron Corporation 100 Royall Street Canton, MA 62021 (617) 828-2500 4200 Rheology Test System (bench-top capillary rheometer) 3210 Capillary Rheometer Accessory (for use with universal testing machine) 3211 Rheological Testing Instrument (capillary rheometer) 3250 Rotational Rheometer (cone-plate with torque and normal force measurement) Kayeness, Inc. Box 30 Honeybrook, PA 19344 (215) 273-3711 Melt Indexer Galaxy V Capillary Rheometer Killion Extruders, Inc. 56 Depot Street Verona, NJ 07044 (201) 239-0200 Laboratory Rheology Extruder 626 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING Metravib ftD.S. BP 182 69132 Ecully Cedex France Viscostrain extensional rheometer Viscoanalyser (compressional oscillation rheometer) Metrilec Sari 20, Rue Michal 75013 Paris France Rheoplast RClO (pre-shear capillary rheometer) Monsanto Instruments & Equipment 2689 Wingate Ave. Akron, OH 44314 (216) 745-1641 MDR2000 Moving Die Rheometer (oscillatory shear cure meter) ODR 2000 Oscillating Disc Rheometer (for curing studies) Capillary rheometer Die swell detector MTS Systems Corporation Box 24012 Minneapolis, MN 55424 (612) 937-4000 Computer Controlled Capillary Rheometer Model 831 Elastomer Test System Nametre Company 101 Forrest Street Edison, NJ 08840 (201) 494-2422 Vibrational Viscometer (for low viscosity liquids) APPENDIX E 627 Polymer Laboratories, Inc. P. O. Box 1581 Stow, OH 44224 (216) 688-7339 Dielectric Thermal Analyser Dynamic Mechanical Thermal Analyser Rheometrics, Inc. One Possumtown Road Piscataway, NJ 08854 (201) 560-8550 Mechanical Spectrometer (cone-plate with torque and normal force measurement) Dynamic Analyzer (small amplitude oscillatory shear) Stress Rheometer (rotational constant torque rheometer) Extensional Rheometer (based on Miinstedt design) Melt Analyzer (for automatic quality control) On-Line Rheometer (concentric cylinder rheometer) Melt Flow Monitor (on-line slit rheometer) Optical Analyzer (optical accessory) Rosand Precision Limited 11 Little Ridge Welwyn Garden City Herts AL720H, England Rosand Automatic Melt Indexer Seiscor Technologies P. O. Box 470580 Tulsa, OK 74147-0580 (918) 252-1578 CMR-II Process Rheometer (on-line capillary rheometer) Flow Characterization Rheometer (on-line capillary rheometer -several shear rates) 628 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING Seiscor/Han Rheometer (on-line slit rheometer to measure exit pressure) AKSOOO Particle Sampling System Shimadzu Scientific Instruments, Inc. 7102 Riverwood Drive Columbia, Maryland 21046 (301) 381-1227 CFT-500 Flowtester (high-stress, weight-driven capillary rheometer) CFT-20 Flowtester (small, weight-driven capillary rheometer) CMD Automatic Mooney Viscometer Testing Machines Inc. 400 Bayview Ave. Amityville, NY 11701 (516) 842-5400 Ray Ran Melt Flow Indexers Mooney Viscometer Time-Temperature Instruments Inc. P. O. Box 40156 Pittsburgh, P A 15201-0156 (412) 621-5009 MECA Creep Rheometer (high-precision constant stress rheometer for creep and recovery) Tinius Olsen Testing Machine Company, Inc. P. O. Box 429 Willow Grove, PA 19090-0429 (215) 675-7100 Extrusion Plastometer (melt indexer) Sieglaff-McKelvey Rheometer (capillary rheometer) Toyo Seiki Seisaku-Sho, Ltd. 15-4, 5-Chome Takinogawa, Kita-Ku Tokyo 114, Japan APPENDIX E 629 Westover Rheometer (very high pressure capillary rheometer) Capirograph (capillary rheometer) Labo-Plastomil (torque rheometer) Melt Indexer Zwick of America, Inc. P. O. Box 997 East Windsor, CT 06088 (203) 623-9475 Extrusion Plastometers Nomenclature NOTE Equation number refers either to the definition of the quantity or to the principal equation in which the quantity appears. Some symbols that are used only once and defined in the text are not listed here. ROMAN LETTERS a Various empirical constants, or length characteristic of chemical structure of molecule (2-95) aT Shift factor for time-temperature superposition A Surface area or interfacial area, or strain scale factor in exponential shear (5-75), or empirical constant (15-3) AG Linear viscoelastic property defined in (2-74) and (2-75) Ai Amplitude of component Ei of the electric vector b Various empirical constants or Rabinowitch correction (8-20) B Extrudate swell ratio for capillary (8-62) or empirical con- stant (15-2, 15-3) Boo Ultimate swell ratio for capillary extrudate B A Area swell ratio of parison (16-4) BD Diameter swell ratio of parison (16-1) BH Thickness swell ratio of parison (16-2) Bw Weight swell ratio of parison (16-3) Bij Component of Finger tensor c Instrument compliance or velocity of light C Stress optical coefficient or constant in Equation 15-5 Co Constants in WLF equation (2-129) Cij Component of Cauchy tensor d Tube diameter in Doi-Edwards model df Flight clearance 630 NOMENCLATURE 631 D Diameter or drop shape parameter (11-6) Db Diameter of barrel D Diameter of the root of the screw Do Diameter of capillary die D(t) Tensile creep compliance D /Dt Substantial derivative 0-59) e Bagley end correction (8-48) or flight width in extruder 04-4) ei Unit normal vector (i = 1, 2 or 3) E Young's modulus 0-12) E Extruder power input 04-7) Ef Contribution of clearance flow to extruder power 04-14) Ea Activation energy for flow (2-128) and 00-15) Ei Component of electric field vector E(t) Tensile relaxation modulus flEoo Constant in Equation 12-5 fa Fractional free volume F Force F(A) Relaxation spectrum function (2-28) Fij Component of displacement gradient tensor (A-4) g Acceleration due to gravity gi Relaxation strength of ith element of model (5-41, 5-44) g/i) Relaxation strength for tensile recoil (6-22) G Shear modulus 0-13) Gi Modulus of ith Maxwell element Gc Crossover modulus (2-78) Gd Amplitude ratio (lTo/Yo) in oscillatory shear Gr Reduced modulus (2-119) G(t) Shear stress relaxation modulus G~ Plateau modulus G'(w) Storage modulus G"( w) Loss modulus G*(w) Complex modulus G~ Apparent value of storage modulus including the effect of instrument compliance G:; Apparent value of loss modulus including the effect of instrument compliance h Plate spacing or rheometer gap hi Damping function of ith model element (5-27) h( ) Damping function 632 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING HOt) Relaxation spectrum function (2-29) H Channel depth H(Il' 12 ) Damping functional (3-82) II First scalar invariant of a Cartesian tensor 12 Second scalar invariant of a Cartesian tensor 13 Third scalar invariant of a Cartesian tensor J Shear compliance Jr Reduced compliance J(t) Shear creep compliance JJ Steady state compliance in limit of very small shear stress J'(w) Storage compliance J"( w) Loss compliance J*(w) Complex compliance k Thermal conductivity of melt K Empirical constant or rheometer geometry constant (7-1, 7-2) Ke Spring constant Kv Dash pot viscous resistance coefficient K", Constant in Equation 12-5 L Length of sample or of capillary or slit or of extruder barrel L Tube length in Doi-Edwards model Lo Initial length of sample m Various empirical constants or parameter for multiaxial extensional flows (6-38) met) Memory function (3-17) M Molecular weight of monodisperse polymer or torque Mi Molecular weight of component i M( ) Strain-dependent memory function (3-61) Me Critical molecular weight for appearance of entanglement effects in the dependence of 7]0 on molecular weight M: Critical molecular weight at which JJ ceases to depend on M Mi Molecular weight of material having weight fraction -w: Mo Monomer molecular weight or torque amplitude in oscilla• tory shear for rotational rheometer or parameter of log• normal molecular weight distribution (B-6) Me Average molecular weight between entanglements Mn Number average molecular weight (see Appendix B) NOMENCLATURE 633 Mw Weight average molecular weight (see Appendix B) M z Z-average molecular weight (see Appendix B) MI Melt index n Various empirical constants or index of refraction n Outer-directed unit normal vector nij Birefringence tensor n' Real component of complex refractive index nil Imaginary component of complex refractive index N Angular screw speed No Avogadro's number N I( y) First normal stress difference (simple shear) Ni y) Second normal stress difference (simple shear) Nit) First normal stress difference during transient shear defor• mation Nit) Second normal stress difference during transient shear deformation N I( t) + First normal stress growth function Nit)+ Second normal stress growth function p Ratio of disperse phase viscosity to matrix viscosity P Pressure, or degree of polymerization, or energy dissipa- tion per cycle per unit volume in oscillatory shear Pa Ambient pressure Pd Driving pressure-capillary flow Pe Exit pressure P* Hole pressure (8-36) Po Extruder feed pressure dP Pressure difference (P2 - PI) dPcap Pressure drop for fully-developed capillary flow dPends Entrance loss plus exit loss (8-47) dPent Entrance pressure drop d Pex Exit pressure drop PI Polydispersity index Q Volumetric flow rate Qd Drag flow rate r Radial spatial coordinate in cylindrical coordinates R Radius of rheometer fixtures or of blown film bubble, or ratio of transmitted light intensity to incident intensity Roo Ultimate recoil coefficient R(t) Recoil function (recoverable compliance) (2-39) R g Radius of gyration of molecule Re Reynolds number (1-62) s (t - t') or interfacial tension S Linear strain or strength of network at the gel point (12-8) or order parameter for liquid crystal polymer (17-1) Soo Ultimate tensile recoil function, Eoo/aE (see Equation 6-19) (Equal to D2 for linear viscoelastic behavior) Sij Nonlinear strain tensor (3-65) t Time te Reentanglement time during interrupted shear (5-45) t E Reentanglement time during reduction in shear rate (5-46) fl.t Rise time in "step" strain experiment ti Component of surface stress vector to A specific time at which strain is introduced into sample tr Reduced time (2-120) or rest time during interrupted shear T Temperature Ta Empirical constant in (15-6) Tg Glass transition temperature (see Appendix D) Tm Melting temperature To Reference temperature or wall temperature (7-4) or con• stant in (15-2) u Time-dependent elastic energy potential function (3-57, 3-58) u i Displacement vector component in Xi direction U Strain-dependent elastic energy potential function (3-59) v Fluid velocity or velocity of light VI Free volume Vi Velocity component in Xi direction v bx Transverse component of relative boundary plane velocity Vbz Down-channel component of relative boundary plane ve- locity vM Melt velocity in machine direction (blown film) V x Transverse component of melt velocity in screw channel vz Down-channel component of melt velocity in screw chan• nel NOMENCLATURE 635 V Velocity of moving plate or dimensionless down-channel velocity Vb Linear velocity of extruder barrel ~ Slip velocity (8-26) Wi Weight fraction of component i W Width of slit or perpendicular distance between flights We Weber number (11-7) W Work per cycle per unit volume of fluid Xi Spatial coordinate or displacement function (A-I) X M Machine direction coordinate (blown film) x T Transverse direction coordinate (blown film) X Reactive group conversion (12-1) d X Plate displacement y Radial coordinate in extruder equations Y Dimensionless radial distance in extruder equations z Distance of fluid element above a horizontal datum plane or axial coordinate GREEK LEITERS a Empirical constant in (3-76), or exponential rate coeffi• cient in exponential shear (5-75), or coefficient of thermal expansion of fluid (D-l), or pressure coefficient of viscosity (10-17), or thermal diffusivity of melt f3 Empirical constant in (3-79), or Rabinowitch correction for slit flow (8-34), or angle between polarization direction and principal direction of refractive index tensor, or modi• fied Brinkman number (14-18), or parameter of log-normal MWD (B-6) 'Y Shear strain 'Yi Step strain introduced at time ti 'Yo Magnitude of shear strain at t = 0 in step-strain experi- ment 'Yo Strain amplitude in oscillatory shear 'Y, Recovered shear strain 'Y2 Ultimate recoil (recoverable shear) for linear viscoelastic behavior 'Yoo Ultimate recoil (recoverable shear) (2-38) 636 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING 'Y5 Shear strain at which (J" is maximum in stress growth 'YN Shear strain at which NI is maximum in stress growth Y Shear rate 'Ym Mean shear rate in superposed steady and oscillatory shear (5-69) YA Apparent wall shear rate (8-13) YA 5 Apparent wall shear rate corrected for wall slip (8-26) Yo Empirical constant with units of shear rate in Equation 15-2 Yw Absolute magnitude of wall shear rate Y* Representative shear rate (8-21) 'Y.ij Component of infinitesimal strain tensor 'Yij Component of rate of deformation tensor r Surface tension of melt ° Mechanical loss angle 0i Phase angle of component Ei of the electric vector 0ij Kronecker delta (3-7) { Monomeric friction coefficient C Hencky strain c/t) Tensile recoil function Coo Ultimate tensile recoil (6-19) Co Magnitude of Hencky strain imposed at t = 0 in exten• sional step strain i Hencky strain rate i M Extensional strain rate in machine direction (blown film) iT Extensional strain rate in transverse direction (blown film) i B Biaxial extensional strain rate (6-24, 6-25) TJ Viscosity [ TJ] Intrinsic viscosity (see Appendix C) TJi Viscosity of ith Maxwell element (Gjlt) TJ( y) Viscosity function TJo Zero shear viscosity TJA Apparent viscosity (8-15) TJ! Viscosity of suspending fluid TJ p Plastic viscosity (1-24) TJOi Zero shear viscosity of component i TJoo Constant in Equation 12-5 TJ +(t) Shear stress growth coefficient NOMENCLATURE 637 17 -(t) Shear stress decay coefficient 17;(0 Tensile stress growth coefficient 17;(t) Biaxial stress growth coefficient 17ii B) Biaxial extensional viscosity 17E( y) Extensional viscosity (uniaxial extension) 7j(i) Extensional viscosity (uniaxial and biaxial extension) 17EC Apparent extensional viscosity determined in entrance flow 17 ES Apparent extensional viscosity determined in melt strength test 17'(W) Dynamic viscosity 17"(W) In-phase component of complex viscosity 17*(W) Complex viscosity 17 e Exponential viscosity (5-80) 17 P Principal exponential viscosity (5-83) 17, Relative viscosity (11-2) and (C-l) ()o Cone angle of cone-plate rheometer fixtures e Angle between molecular axis and the direction of a liquid crystal polymer or helix angle of screw flights or angle between tangent to bubble curve, R(z) curve, and vertical (z) axis, i.e., arctan dR/dz A Relaxation time or other characteristic time of fluid or wavelength of oscillation of electric vector Ae Equilibration time in Doi-Edwards model Ad Diffusion time in Doi-Edwards model Ao Characteristic time of a material (a material property) Al Terminal (longest) relaxation time AR Longest relaxation time in the Rouse model (2-98) Ai Relaxation time of ith Maxwell element As Shear wavelength (7-10) P Melt density Po Melt density at the reference temperature (I' Shear stress in simple shear (l'w Absolute magnitude of wall shear stress (l'B Net stretching stress, biaxial extension (l'E Principal stretching stress (1-34) (I'm Maximum stress during interrupted shear experiment (l'min Minimum stress during reduction of shear rate experiment (1'0 Stress amplitude in oscillatory shear or yield stress (l'ij Component of the stress tensor 638 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING 0" + (t) Shear stress growth function O"(t, 'Y) Shear stress relaxation function 'Tij Component of extra stress tensor c/> Volume fraction of filler c/>m Maximum packing fraction of filler C/>O Angular amplitude for oscillatory shear in a rotational rheometer Dimensionless pressure in extruder X Extinction angle 'l'1( y) First normal stress coefficient 'l'i y) Second normal stress coefficient "1'1,0 Limiting value of '1'1 as y ~ 0 '1'2,0 Limiting value of '1'2 as y ~ 0 "I't(t) First normal stress growth coefficient "I';(t) Second normal stress growth coefficient '1'1-(1) First normal stress decay coefficient '1'2-(1) Second normal stress decay coefficient w Frequency n Rotational speed (rad/s) Author Index Numbers shown in italics refer to reference lists at ends of chapters. Abbott, T. N. G., 284, 295 Balke, S., 607, 612 Acierno, D., 290, 296 Ballenger, T. F., 336, 338, 339, 344 Adams, E. B., 137, 140, 152 Ballman, R. L., 249, 267,551, 556 Adams, N., 282, 295 Balta Calleja F. J., 498, 508 Adolf, D., 420, 423 Bartels, CR., 166, 177 Agarwal, P. K., 208, 212, 213, 229 Bashara, N. M., 352, 354, 363 Agassant, J. F., 329, 343,499, 508, 538, 555, Basu, S., 520, 521, 529, 530 595,600 Bata, G. L., 407, 409 Agrawal, A. R., 565, 566 Batchelor, G. K., 275, 295 Ajroldi, G., 519, 529 Baumgartel, B., 72, 101 Akana, Y., 552, 556 Bayer, R. K., 247, 267,498, 508 Aldhouse, S. T. E., 319, 342 Bennett, K. E., 249, 262, 267 Allan, P. S., 498, 508 Bentley, M. E., 517, 529 Allen, W. F., 553, 556 Berardinelli, F. M., 523, 530 Amari, T., 411, 421 Bergem, N., 338, 344 Anders, S., 362, 364 Bernstein, B., 127, 151 Anderson, R. D., 411, 413, 415, 416, 421 Berry, G. C, 78, 101, 164, 177, 208, 209, Andrade, E. N. de C, 290, 296 229, 426, 439 Andrews, R. D., 99, 102, 410, 421 Bessho, N., 190, 229 Anturkar, N. R., 554, 556 Bevis, M. J., 498, 508 Aris, R., 25, 41 Beyer, C E., 517, 529 Arman, J., 63, 69, 100, 101, 102 Biesenberger, J. A., 415, 419, 422, 445, 484, Armstrong, R. C, 27, 41, 85,101,107,151, 485,490 156, 161, 163, 176, 177, 346, 363 Billingham, N. C, 615, 616 Asada, T., 359, 364, 431, 439 Billmeyer, Jr., F. W., 613, 616 Ashare, E., 171, 172, 177,222,230,276,295 Binding, D. D., 324, 343 Astarita, G., 222, 230 Binnington, R. J., 346, 363 Athey, A. J., 539, 555 Bird, R. B., 27, 41, 85, 101, 107, 151, 156, Au-Yeung, V. S., 518,529 157,161,176,225,230,313,314,342,360, Awaya, H., 339, 344 361, 364 Azzam, R. M. A., 352, 354, 363 Blacklock, J. E., 539, 555 Blake, J. W., 411, 413, 415, 416, 421 Blakeslee, III, T. R., 539, 555 Baba, S. M., 475, 489 Blow, M. M. J., 417, 423 Bagley, E. B., 320, 342 Boger, D. V., 314, 342, 346, 362, 363 Baird, D. G., 311, 312, 315, 318, 324, 342, Bogue, D. C, 137, 140, 152, 331, 336, 338, 343, 362, 364, 436, 437, 440, 495, 508 339, 343, 344, 359, 360, 363, 364 Bakerdjian, Z., 334, 343, 434, 440, 584, 600 Boiko, B. B., 338, 344 Balin, P. L., 471, 489 Boles, R. L., 331, 343 639 640 AUTHOR INDEX Booij, H. c., 60, 61, 66, 100, 130, 133, 137, Christiansen, E. B., l70, 177 139,151,152,174,175,177,178,219,229 Chu, E., 495, 499, 500, 507, 508 Booy, M. L., 453, 465, 466, 489 Chung, C. I., 477, 489 Borisenkova, E. K., 338, 344 Chung, S. c.-K., 265, 268 Bormuth, H., 506, 508 Cieloszyk, G. S., 537, 555 Boukhili, R., 495, 507 Clark, E. S., 172, 177,400,409,497,508 Bowers III, G. H., 399, 409 Clegg, D. W., 270, 275, 294 Brenna, A., 282, 295 Cline, A. W., 536, 555 Brewer, G. W., 496, 508 Co, A., 554, 556 Bright, P. F., 399, 409 Cogswell, F. N., 242, 249, 258, 260, 266, 267, Brizitsky, V. I., 332, 335, 343 322, 338, 343, 344, 386, 389, 401, 409, Broadhead, T. 0., 562, 566 435-438,440,515,516,518,520,526,529, Brodkey, R. S., 274, 294 538, 555, 598, 600 Brown, R. A., 346, 363 Cohen, R. E., 163, 177 Bruker, I., 194, 229,281,283, 295 Colaluca, M. A., 414, 422 Bueche, F., 75, 101, 162, 177,378, 389 Cole, K. S., 90, 101 Burkert, S. J., 414, 415, 422 Cole, R. H., 90, 101 Burkhardt, U., 486, 487, 490 Collyer, A. A., 270, 275, 294 Busse, W. F., 247, 267, 399, 409 Connelly, R. W., 221, 230 Butcher, A. F., 203, 204, 229 Cook, K. S., 294, 297 Cooper, A. R., 607, 612 Cooper, S. L., 157, 176 Cain, J. J., 550, 556 Corbett, H. 0., 547, 555 Cakmak, M., 523, 530 Cox, R. G., 403, 404, 409 Calundann, G. W., 425, 439 Cox, R. H., 174, 177 Campbell, G. A., 544, 546, 547, 555 Cox, W. P., 173, 174, 177 Cancio, L. V., 471, 489, 517, 529 Crady, D. L., 264, 268 Cantow, M. J. R., 374, 375, 389 Cragg, L. H., 613, 616 Cao, B., 546, 555 Crawley, R. L., 65, 100, 195-198, 229,282, Carley, J. F., 477, 489 295, 334, 343 Carr, S. H., 406, 409 Crist, B., 166, 177 Carreau, P. J., 162, 177 Crochet, M. J., 37, 41, 335, 344 Carroll, D. R., 414, 422 Cross, M. M., 162, 177, 283, 295, 378, 389 Castro, J. M., 411, 416, 421 Crothers, D. M., 614, 616 Cat ani, A. M., 407, 409 Crowder, J. W., 336, 338, 339, 344 Chambon, F., 420, 423 Crowson, R. J., 399, 400, 409 Chan, D., 273, 274 Crozier, D., 419, 423 Chan, H. L. W., 416, 422 Currie, P. K., 149, 152 Chan, Y., 400, 405, 409 Curry, J., 565, 566 Chapman, F. M., 397, 400, 408 Curtiss, C. F., 27, 41, 85, 101, 107, 151 Chapman, G. R., 539, 555 Czarnecki, L., 400, 409 Charles, M., 320, 333, 343 Chatraei, Sh., 262, 267 Chattopadhyay, S., 159, 177, 328, 329, 343 D'yakov, K. D., 561, 565 Chella, R., 483, 490 Dannhauser, W. c., 90, 91, 93, 101 Chen, I-Jen., 336, 338, 339, 344 Darby, J. P., 510, 529 Chen, K.-c., 332, 343 Darby, R., 290, 296 Chen, S. c., 493, 507 Darnell, W. H., 472, 489 Cheng, D. C.-H., 390, 391, 408 Davies, J. M., 286, 296, 313, 342 Chiang, H. H., 163, 177, 505, 508 Davis, H. L., 331, 343 Chmiel, H., 304, 326, 341 Davison, S., 406, 409 Chou, C. H., 317, 342 deCindio, B., 290, 296 Chow, A. W., 276, 295 de la Lande, M., 499, 508 Choy, I.-C., 415, 422 De Rossett, T. A., 334, 343 Christiano, J. P., 536, 555 de Vargas, L., 312, 342 AUTHOR INDEX 641 Dealy, J. M., 150, 151, 152, 204-206, 219, Ericksen, J. L., 426, 427, 439 223-228, 229, 230,241,244,260,261,265, Erwin, L., 483, 484, 490, 521, 530, 553, 554, 266-268,270,285-287,290,292-294, 294, 556 296,297,326-329,334,343,414,422,511, Everage, A E., 249, 267 514,516-519,521, 529,554,555,557,562, 565, 566, 570, 571, 599 Dearborn, J. R., 519, 521, 522, 529 Fan, X. J., 225, 230 Degeneuve, G., 99, 508 Farber, R., 544, 555 deGennes, P. D., 81, 101,424,426,427,430, Faucher, J. A, 294, 297 439 Fayt, R., 405, 409 DeKee, D., 176, 178 Fellers, J. F., 359, 364 Delos, S., 417, 422 Fenner, R. T., 442, 489 Demarmels, A, 151, 152,253,264-266, 267, Fernandez, D., 315, 342 268 Fernandez, F., 520, 521, 529, 530 Demus, D., 424, 439 Ferry, J. D., 33n, 41, 43, 50, 55, 64, 65, 75, Denn, M. M., 174, 177, 307, 314, 336, 338, 78,80,89,91,94, 100, 101, 219, 229,382, 339, 340, 342, 344, 550, 554, 556 389, 584, 600 Denson, CD., 261, 264, 267, 268 Fetters, L. J., 166, 177, 357, 362, 364 Derezinski, S. J., 493, 507 Fifer, R. L., 523, 530 Dierckes, A C, 158, 177 Figlan, J., 411, 421 Dijksman, J. F., 495, 507 Finger, F. L., 336, 344 Dillon, J. G., 614, 616 Finlayson, B. A, 313-315, 342 Dillon, J. H., 410, 421 Fisa, R., 495, 507 Dillon, R. E., 339, 344 Fischer, E., 519, 529 Dimitrov, M., 471, 489 Fleissner, M., 290, 296 Dobroth, T., 553, 556 Flory, P. J., 411, 421, 425, 439, 610, 612, 619, Dodge, J. S., 285, 295 621 Doi, M., 79, 82, 83, 85, 101, 107, 146-148, Flumerfelt, R. W., 157, 176 151,349,357,359,362,363,364,426,439 Folkes, M. J., 399-401, 409 Donis, R., 399, 409 Folt, V. L., 414, 422 Dormier, E. J., 205, 229, 511, 529 Fortuin, J. M. H., 279, 295 Doshi, S. R., 150, 152, 225-228, 230 Fox, T. G., 78, 101, 164, 177,619,621 Dreiblatt, A, 565, 566 France, G. H., 327, 343 Drexler, L. H., 313, 319, 342,354,356,357, Franck, A, 258, 259, 267, 286, 296 363 Frattini, P. L., 357, 363 Du, C-C, 538, 555 Freeman, W., 417, 422 Dusi, M. R., 411, 413, 417, 419, 421-423 Friedman, E. M., 593, 600 Dutta, A, 521, 529 Friedrich, C, 175, 178 Dynes, P. J., 420, 423 Fritch, L. W., 505, 508 Fritz, H. G., 510, 529, 560, 565, 565, 566 Fritzen, J. S., 411, 413, 417, 421 Froelich, D., 245, 267 Edelman, R., 523, 530 Fujiki, T., 388, 389, 540, 555 Edwards, R., 554, 556 Fujiyama, M., 339, 344 Edwards, S. F., 81-83, 101, 107, 146-148, Fukada, M., 186, 188, 189, 196, 199, 228, 229 151,359,362,364,370, 389 Fuller, G. G., 276, 295, 345-347, 360, 362, Einaga, Y., 188, 229 363,364 Eise, K., 486, 487, 490 Furches, B. J., 506, 508 Eisenberg, A, 618, 621 Furumiya, A, 552, 556 EI Kissi, N., 318, 340, 342 Elbirli, B., 163, 177, 475, 489 Eley, R. R., 411, 421 Ganani, E., 283, 295 Elia, A E., 498, 508 Garbella, R. W., 283, 295 Elmendorp, J. J., 405, 406, 409 Garcia, M. N., 414, 422 Enns, J., 411-413, 421 Garcia-Rejon, A, 334, 343, 514, 516-518, Ente, J. J. S. M., 552, 556 529 642 AUTHOR INDEX Garritano, R., 415, 419, 422 Hagnauer, G. L., 413, 421 Garverick, S., 417, 422 Halasz, L., 563, 566 Gaskins, F. H., 323, 343 Hale, A, 414, 422 Gates, P. c., 538, 555 Han, C. D., 90, 101, 173, 177,313-316,319, Gauvin, R., 495, 507 320,333,342,354,356,357,363,390,397, Gavin, P. T., 280, 295 400,402,408,409,411,415,419,421,423, Gavins, J., 332, 343 437, 438, 440, 549, 550, 556 Geiger, K, 283, 295 Hannon, M. J., 411, 421, 518, 521, 525,529, Gendron, R., 174, 177 590-593, 600 Gent, AN., 290, 296 Hansen, M. G., 281, 295 George, H. H., 363, 364 Hanson, D. E., 205, 229 Gergen, W. P., 406, 409 Harban, A A, 571, 599 Ghijsels, A, 407, 409, 552, 556,584, 600 Harding, S. W., 162, 177, 378, 389 Giacomin, A J., 290, 292, 293, 296, 297,562, Harran, D., 415, 420, 422 566 Hashimoto, N., 488, 490 Gibson, A. G., 331, 343 Hassager, 0., 27, 41, 85, 101, 107, 151, 156, Gillham, J. K, 411-413, 416, 421, 422 161, 176 Githuku, D., 248, 267 Hatz, R., 424, 439 Glasscock, S. D., 334, 343 Hauson, E. E., 410, 421 Gleissle, W., 175, 178, 280, 282, 294, 295 Hayashi, T., 497, 508 Glenn, Jr., W. B., 493, 507 Hedvig, P., 417, 422 Godfrey, J., 417, 422 Hegele, R., 471, 489 Goettler, L. A, 399, 409 Heinle, P. J., 414, 421 Gogos, C. G., 37, 41, 158, 177,203, 229,332, Heinz, W., 561, 565 343, 442, 488, 489, 493, 507 Helfand, 85, 101,225, 230 Goldfarb, I. J., 416, 422 Henderson, AM., 335, 344, 514, 529 Goldstein, c., 285, 295 Henry, R. L., 290, 296 Gonzalez, H., 521, 530 Henze, E. D., 516, 529 Gonzalez-Romero, V., 415, 416, 418-420, Herrington, F. J., 547, 555 422 Herrmann, H., 486, 487, 490, 565, 566 Gotro, J. T., 411, 417, 419, 421, 422 Hess, J. E., 414, 421 Gotsis, A D., 318, 324, 342, 436, 437, 440 Heuser, G., 279, 295 Gattfert, A, 559, 565 Hieber, C. A, 163, 177, 505, 508 Gottgetreu, S. R., 475, 489 Higashitani, K, 311, 342 Gottlieb, M., 285, 295, 313, 314, 342 Hill, Y., 290, 296 Gouz, J. J., 503, 508 Hilton, D. c., 264, 268 Goyal, S. K, 499, 500, 508 Hinch, E. J., 360, 364 Graessley, W. W., 65, 80, 84, 85, 88, 95, Hinrichs, R. J., 411, 421 100-102, 166, 177, 182, 187, 190, 192, 193, Hoff, M., 417, 422 195-199,201,202,228,229,282,295,334, Hoffman, D. M., 411, 415, 417, 421 343, 369, 370, 373, 377-379, 386, 389 Hoftyzer, P. J., 370, 383, 384, 389, 619, 621 Grattom, R. F., 286, 296 Holmes, L. A, 171, 172, 177 Greener, J., 221, 230 Honerkamp, J., 72, 101 Gregory, D. R., 384, 389 Hong, C.-N., 317, 332, 342, 343 Greygang, G. G., 503, 508 Horie, K, 419, 423 Griffin, A X., 570, 599 Horrion, J., 577, 600 Griffith, R. M., 335, 344 Hostettler, J., 283, 295 Gross, L. H., 285, 295 Hou, T. H., 419, 423 Gulrich, L. W., 425, 439 Hoverty, P., 417, 422 Gupta, R. K, 544, 555 Hrymak, AN., 500, 508 Gupta, V. B., 482, 489 Hsu, C. c., 158, 177, 493, 507 Hsu, T., 249, 262, 267 Hadad, D. W., 411, 413, 417, 421 Huang, D. c., 335, 344 Haessley, W. P., 522, 530 Huang, T., 544, 547, 555 Haghtalab, A, 292, 297 Huilgol, R. R., 150, 152 Hagler, G. E., 336, 338, 339, 344 Hull, AM., 505, 508, 587, 600 AUTHOR INDEX 643 Huppler, J. D., 171, 172, 177 Kemp, R. A, 283, 295 Hiirlimann, H. P., 227, 230, 290, 291, 293, Kepes, A, 561, 565 296 Kerker, M., 349, 363, 424, 439 Hushower, M. E., 419, 423 Keunings, R., 37, 41 Hutton, J. F., 281, 295,313, 342 Khachatryan, G. M., 561, 565 Khan, S. A, 132, 152, 152, 390, 408 Ide, Y., 258, 259, 267, 435, 438, 440 Khanna, Y. P., 166, 177 Ingen Housz, J. F., 475, 489 Khatta, R. K, 417, 422 Inomiya, K, 78, 101 Kietz, W., 497, 508 Inoue, T., 97, 98, 102 Kim, H. T., 510, 529 Insarova, N. I., 338, 344 Kim, J. T., 471, 489 Isayev, A I., 222, 230, 332, 335, 343 Kim, S.-G., 495, 508 Isherwood, D. P., 471, 489 Kimura, M., 576, 600 Isono, Y., 219, 229 Kimura, S., 188, 190-193, 229, 290-292,296 Iwama, M., 615, 616 King, R. G., 402, 409 Iyengar, V., 554, 556 Kiss, A D., 357, 362, 364 Kiss, G., 359, 364, 435, 436, 440 Jabarin, S. A, 523, 530, 579, 600 Kitagawa, K, 95-97, 102 Jackson, S., 565, 566 Kitano, T., 393, 397, 408 Jaffe, M., 425, 439 Klein, I., 442, 447, 460, 467, 473, 475, 479, Jakopin, S., 486, 487, 490 489 Janeschitz-Kriegl, H., 140, 152, 262, 267, Knapper, K M., 284, 295 292, 297, 358, 359, 362, 364, 495, 507 Kojima, C. J., 419, 423 Janssen, L. P. B. M., 442, 487, 489 Kojima, T., 220, 230 Jenkins, J. T., 265, 268 Kojimoto, T., 190, 229 Jerome, R., 405, 409, 577, 600 Kolinski, A, 86, 101 Johnson, J. F., 374, 375, 389 Komatsubara, T., 503, 508 Jones, D. N., 547, 551, 555 Kondo, A, 318, 342 Jones, L. G., 274, 294 Koopmans, R. J., 336, 344, 515, 529 Jones, T. E. R., 286, 296 Kotaka, T., 219, 229 Jong, W. R., 505, 508 Kouba, K, 538, 555 Jongschaap, R. J. J., 222, 230, 284, 295 Kozicki, W., 317, 342 Joseph, D. D., 276, 295 Kral, V., 538, 555 Joyner, R. S., 471, 489, 517, 529 Kranbuehl, D., 417, 422 Jud, K, 495, 507 Krause, E., 279, 295 Krauskoff, L. G., 324, 343 Kachin, G. A, 506, 508 Krieger, I. M., 285, 295, 296, 397, 408 Kaiser, H., 519, 529 Kruder, G. A, 471, 489 Kalika, D. S., 174, 177, 307, 338, 339, 340 Krul, N., 520, 529 Kahenbacher, E. J., 554, 556 Kumar, N. G., 160, 177 Kalyon, D., 517, 518, 521, 529, 538, 555 Kumar, R., 166, 177,415, 419, 422 Kamada, K, 615, 616 Kuo, C C, 416, 422 Kamal, M. R., 334, 343, 362, 364, 407, 409, Kurata, M., 57, 100, 183-188, 193, 195, 196, 415,418,422,423,434,440,495,500, 507, 199, 200, 219, 228, 229, 290, 291, 292, 508, 517, 518, 521, 529, 584, 600 296, 615, 616 Kambe, H., 419, 423 Kurtz, S. J., 334, 343, 537, 539, 547, 551, 554, Kanai, T., 547, 555 555,556 Kataoka, T., 393, 397, 400, 408, 409 Kuzuu, N. Y., 85, 101, 426, 439 Katsaros, J. D., 406, 409 Kwack, T. H., 549, 556 Katsyutsevich, E. V., 222, 230 Kwolek, S. L., 425, 439 Kausch, H. H., 495, 507 Kwon, T. H., 331, 343 Kaye, A., 127n, 283, 295 Kearsley, E. A, 127, 151 Labana, S. S., 421, 423 Keenan, J. D., 419, 423 Lafleur, P. G., 495, 507 Keentok, M., 281, 295 LaLiberte, B. R., 413, 421 Kemblowski, Z., 410, 421 La Mantia, F. P., 290, 296 644 AUTHOR INDEX Lambright, A. J., 399, 409 Ma, C.-Y., 395, 397, 398, 400, 408 Lane, J. W., 417, 422 Maalcke, R. J., 405, 409 Larson R. G., 54, 100, 107, 149, 151, 132, Macdonald, I. F., 222, 230, 276, 295 137,143,145,149-151,152,175,178,227, Mackay, M. E., 346, 362, 363 230, 238, 265, 266, 268, 430, 439 Mackley, M. R., 319, 342 Latrobe, A., 499, 508 MacKnight, W. J., 416, 422 Laudouard, A., 415, 420, 422 Macosko, C. W., 71,101,140,152,249,262, Laun, H. M., 71, 73, 92, 99, 100, 101, 102, 264, 267, 268, 281, 285, 295,411,413-416, 122,134,135,140,151,152,171,176,178, 418-420,422,423,518,529,561,564,566 183,188,190,198,202,208-210,213, MacSporran, W. c., 283, 295 215-217,228,229,243,244,247-259,264, Maddock, B. H., 473, 477, 489 265, 266, 267, 273, 289-291, 294, 296, Malguarnera, S. c., 414, 422,495, 508 309,315,322-324,327,332,340,341,343, Malkin, A. Ya., 166, 169, 172, 177,419,423 549, 556, 598, 600 Malone, M. F., 406, 409 Laurence, R. L., 249, 262, 267 Mamada, A., 415,422 Lawler, J. V., 346,363 Manaresi, P., 615, 616 Lawton, E. L., 577, 600 Manisoli, A., 495, 508 Leaderman, H., 78, 101, 372, 389 Manzione, L. T., 411, 414, 421, 422 Leal, L. G., 346, 360, 363, 364, 404, 409 Marco, c., 577, 600 Leblans, P., 65,100,133,137,139,152,174, Marianucci, E., 615, 616 175, 177, 178,248, 267 Marin, G., 63, 69, 95,100, 101, 102 Lee, B. L., 414, 422 Maron, S. H., 393, 408 Lee, C. Y.-c., 55, 100,416, 422 Marrucci, G., 432, 439 Lee, D-S., 419, 423 Marsh, B. C., 276, 295 Lee, K. H., 274, 294 Marsh, B. D., 222, 230 Lee, L. J., 335, 344 Martin, G. c., 411, 419, 421 Lee, T. S., 397, 400, 408 Martin, J. E., 420,423 Lee, W. A., 619, 621 Maskell, S. G., 515, 526, 529 Lee, W. K., 598, 600 Mason, S. G., 403, 404, 409 Leib, R. I., 399, 409 Masuda, T., 95-98, 102, 220, 223, 230, 285, Lem, K-W., 90,101,173, 177,411,415,421 296, 552, 556 Lenz, R. W., 576, 600 Matsumoto, T., 223, 230, 285, 296, 359, 360, Leslie, F. M., 426, 427, 439 363,364 Lin, C. M., 554, 556 Mavridis, H., 500, 508 Lin, S. H., 158, 177 Maximovich, M. G., 411, 413, 417, 421 Lin, Y. H., 149, 152 Maxwell, B., 205, 215, 229, 284, 286, 294, Lindt, J. T., 473-475, 489 295-297, 511, 529 Link, G., 92-94, 102, 286, 296 May, C. A., 410, 411, 413, 417, 419, 421, 423 Linster, J. J., 227, 230, 258, 259, 267 Mazich, K. A., 406, 409 Lipshitz, S. D., 419, 423 McCarthy, R. V., 294, 297 Liu, T. Y., 290, 291, 296 McChesney, C. E., 425, 438, 439, 440 Liu, T.-J., 317, 332, 342, 343 McGlamery, R. M., 571, 599 Lobe, V. M., 397, 408 McIntyre, L. V., 225, 230 Lockyer, M. A., 271, 294 McKelvey, J. M., 450, 469, 477, 481, 482, 489 Lodge, A. S., 115, 117, 119, 151, 154, 176, McKenna, G. B., 219, 229, 282, 295 224,230,281-283,295,310,312,313,342 McLeish, T. C. B., 85, 101 Lofgren, E. A., 523, 530, 579, 600 McNally, D., 438, 440, 495, 507 Lopez Cabarcos, E., 498, 508 Mead, D. W., 290, 291, 296 Lopulissa, J. S., 284, 295 Meijer, H. E. H., 475, 489 Lord, H. A., 493, 507 Meins, W., 498, 508 Lorntson, J. M., 518, 529 Meissner, J., 99, 102, 119, 134, 138, 140, 142, Lucchesi, P. J., 554, 556 151, 152, 161, 177, 181, 198, 199, 208, Luenberger, D. G., 480, 489 212-215,227,228-230,243,244,247,249, Luo, X-L., 189, 229 250,253,256-258,262,264-266,266-268, Lupton, J. M., 338, 339, 344 273,282,283,289-291,293,294-296,548, Lutz, R. G., 406, 409 556 AUTHOR INDEX 645 Mendelson, R. A., 336, 344 Nam, S., 539, 555 Menezes, E. V., 199, 201, 202 Narain, A., 276, 295 Merz, E. H., 173, 174, 177 Nazem, F., 281, 290, 295, 296 Metzner, A. 8., 390, 394, 400, 408, 426, 439, Nelson, B., 562, 566 544,555 Nettelnbreker, H.-J., 565, 566 Michaeli, W., 332, 343 Nichols, R. J., 487, 490 Michele, J., 399, 409 Nielsen, L. E., 390, 394, 400, 408 Middleman, S., 37, 41, 158, 176,327, 343, Nishijima, K., 393, 397, 400, 408, 409 447, 467, 469, 489 Nishizawa, K., 183, 184, 195, 228 Mihara, S., 540, 555 Mikkelsen, K. J., 357, 364 Obron, S. J., 70, 101 Mikols, W. J., 416, 422 Oda, K., 172, 177,400,409,410,421,497, Miller, B., 317, 342, 394, 408 508 Miller, J. c., 518, 529, 537, 555 Ohta, S., 189, 196, 199, 220, 229, 230 Min, K., 390, 408, 395, 397, 398, 400, 408 Olbricht, W. L., 404, 409 Minagawa, N., 395-397, 400, 408 Onogi, S., 95-98, 102, 223, 230, 285, 296, Minnick, L. A., 174, 177 359, 364, 431, 439 Minoshima, W., 386, 389,550, 556 Onsager, L., 425, 439 Mita, I., 419, 423 Mitsoulis, E., 314, 342 Onuki, A., 349, 363 Ophir, Z., 435, 438, Mittal, R. K., 482, 489 440 Mochimaru, Y., 276, 295 Orbey, N., 334, 343,515-519, 521, 529 Modan, M., 332, 343 Orchard, S. E., 411, 421 Mokhtarian, F., 484, 490 Osaki, K., 135, 152, 183, 184, 186-193, 195, Mol, E. A. J., 472, 489 196, 199,219, 228, 229,290,291, 292, 296 Mondvai, I., 563, 566 Otsubo, Y., 411, 421 Monge, Ph., 63, 69, 100, 101, 102 Ottino, J. M., 483, 490 Montfort, J. P., 63, 69, 100, 101, 102 Oyanagi, Y., 78, 101, 336, 344, 393, 397, 400, Moon, T. J., 227, 230, 290, 291, 296 405, 408, 409 Mooney, M., 567, 599 Moore, I, P. T., 319, 342 Palmen, J. H. M., 61, 100, 130, 137, 151, Morgan, P. W., 425, 439 152, 174, 175, 177 Morganelli, P., 420, 423 Pan dalai, K., 274, 294 Morgon, B. T., 517, 529 Pandelidis, I. 0., 565, 566 Morikita, N., 497,508 Papanastasiou, A. C., 71,101,140,152,262, Morris, V. L., 419, 423 267 Morse, D. J., 281, 295 Pappas, L. G., 414, 422 Muller, R., 245, 267 Park, J. Y., 550, 556 Muller, S. J., 346, 363 Park, W. S., 65, 100, 195-198, 229 Munari, A., 615, 616 Parnaby, J., 510, 529 Muni, K., 411, 421 Patel, P. D., 68, 100 Miinstedt, H., 242-244, 250-259, 266, 267, Patzold, R., 399, 409 540,555 Payvar, P., 284, 295 Murakami, K., 188, 229, 410, 421 Pearce, E. M., 437, 438, 440 Muramatsu, H., 359, 364 Pearce, P. J., 413, 421 Murayama, T., 290, 296 Pearson, D. S., 85, 101,225, 230,280, 295, Mussatti, F. G., 418, 421, 423 357, 362, 364 Mutel, A., 390, 408 Pearson, J. R. A., 330, 343 Myers, A. W., 294, 297 Pecht, M., 565, 566 Penwell, R. c., 327, 343 Nabata, Y., 415, 422 Perdikoulias, J., 538, 555 Nadkarni, V. M., 159, 177, 328, 329, 343 Perez, G., 388, 389 Nagatsuka, Y., 393, 408 Perry, S. J., 411, 416, 421 Nakajima, A., 552, 556 Petrie, C. J. S., 137, 152,241, 266,336, 344, Nakamichi, T., 411, 421 550, 554, 556 Nakamura, K., 208, 209, 229 Phan-Thien, N., 312, 342 646 AUTHOR INDEX Philipon, S., 499, 508 Riggs, J. P., 425, 439 Philippoff, W., 227, 230, 289, 296, 320, 323, Ritzau, G., 205, 229 333,343 Rivlin, R. S., 290, 296 Piau, J. M., 318, 340, 342 Roberts, E. H., 554, 556 Pierce, P. E., 393, 408 Rochefort, W. E., 225, 230, 280, 295 Pie rick, M. W., 157, 176 Roger, M. G., 336, 344 Pieries, R. N., 471, 489 Rogers, Y. G., 55, 100 Pike, R. D., 311, 312, 315, 342 Rohn, C. L., 92, 102, 561, 564, 565 Pilati, F., 615, 616 Rokudai, M., 388, 389, 540, 555 Pipkin, A. C., 220, 230, 311, 342 Roller, M. B., 411, 414, 419,421,423 Pisipati, R., 495, 508 Roman, J. F., 334, 343 Plazek, D. J., 70, 92,101,102,208,212,213, Roovers, J., 85, 101 229, 286, 296, 415, 422 Rose, W., 495, 507 Plochocki, A. P., 406, 409 Rouse, Jr., P. E., 74, 101 Podolsky, Y. Y., 332, 335, 343 Roxbury, M. L., 290, 296 Pohl, H. A., 203, 229 Roylance, M. E., 413, 421 Pollack, A., 521, 530 Rubin, I. I., 503, 508 Porter, R. S., 327, 343, 359, 364, 374, 375, Rudin, A., 205, 229, 335, 344, 511, 514, 529, 389, 576, 593, 600 539, 555, 570, 578, 599 Portman, P., 264, 265, 268 Rumscheidt, F. D., 403, 409 Powell, R. L., 273, 274, 283, 295 Rutherford, R. A., 619, 621 Prasadarao, M., 437, 438, 440 Ryan, M. E., 415, 422, 495, 507, 521, 522, Prettyman, I. B., 410, 421 529 Prichard, J. H., 388, 389 Ryskin, G., 360, 364 Prilutski, G. M., 426, 439 Pritchard, W. G., 311, 342 Saillard, P., 538, 555, 595, 600 Pritchatt, R. J., 510, 529 Saini, D. R., 159, 177, 328, 329, 343 Proctor, B., 537, 538, 555 Sakai, T., 488, 490 Prud'homme, R. K., 390, 408 Salee, G., 576, 600 Sampers, J., 133, 137, 139, 152, 175, 178, Quinzani, L. M., 280, 295 248, 267 Samurkas, T., 151, 152, 227, 230, 265, 268, 293, 297, 562, 566 Raadsen, J., 407, 409, 552, 556, 584, 600 Sandford, c., 397, 400, 409 Raghupathi, N., 70, 101 Sasahara, M., 393, 397, 408 Raible, T., 253, 262, 265, 267 Saunders, S. W., 290, 296 Rallison, J. M., 346, 363, 404, 409 Sawan, S. P., 411, 421 Ramachandran, S., 170, 177 Sawyers, K. N., 114, 151 Ramamurthy, A. Y., 306, 338-340,341,344, Scarola, L. S., 537, 539, 555 511, 529, 539, 555, 595, 600 Schaefgen, J. R., 425, 439 Ramanathan, R., 324, 343 Schaul, J. S., 518, 521, 525, 529, 590-593, Ramirez, H., 223, 230 600 Rauwendaal, c., 315, 342, 441, 443-445, 460, Schmid, H., 362, 364 461,464,466,467,470,475,477,484,488, Schmidt, R. L., 346, 363 489, 490, 520, 529, 538, 555 Schmitz, K. P., 538, 555 Read, M. D., 311, 312, 315, 342 Schneider, N. S., 416, 422 Read, W. T., 290, 295 Schowalter, W. B., 285, 295 Reddy, J. N., 312, 342 Schreiber, H. P., 205, 229, 511, 529, 570, Reddy, K. R., 313, 332, 335, 342, 343 578, 599 Regester, R. W., 338, 339, 344 Schuch, H., 69, 101,247-249,258,259,264, Revesz, H., 563, 566 265,267,322-324,327,332,343,549,556, Richter, L., 424, 439 598,600 Rhum, D., 411, 421 Schuler, A. N., 576, 600 Rice, P. D. R., 515, 526, 529 Schulken, R. M., 174, 177 Richardson, S. M., 503, 505, 508, 587, 600 Schiimmer, P., 304, 326, 341 Richter, E. B., 411, 421 Schwartz, W. H., 283, 295 Ricketoson, R. c., 505, 508 Schwarzl, F. R., 92-94, 102, 286, 296 AUTHOR INDEX 647 Scriven, L. E., 71, 101, 140, 152 262 264 Strel'tsov, A A, 561, 565 267,268 ' , , Struglinski, M. J., 166, 177 Sebastion, D. H., 519, 521, 522, 529 Sugeng, F., 312, 342 Secor, R B., 264, 268 Suh, N. P., 495, 508 Seferis, J. c., 416, 419, 422, 423, 498, 508 Sundstrom, D. W., 414, 415, 422 Segawa, Y., 223, 230, 285, 296 Swerdlow, M., 520, 529 Selopranoto, J. H., 505, 508 587 600 Semjonow, V., 169, 177 ' , Tadkmor, Z., 37, 41,158,177,332,343,442, Senich, G. A, 416, 422 447,460,467,473,475,479,488 489 493 Senturia, S., 417, 422 495,507 ' , , Serrano, S., 415, 422 Tajima, Y. A, 419, 423 Seth, B. J., 130, 151 Takatori, E., 186, 228 Seto, S., 414, 422 Takigawa, T., 220, 230 Shah, B. H., 290, 296 Tamura, M., 188, 219, 229 Sharma, P. K, 482, 489 Tan, V., 362, 364,407,409,517,529 Shaw, M. T., 163, 177, 334, 343, 414, 422 Tanaka, H., 397, 400, 408, 409 Shen, E. F., 331, 343 Tanner, R I., 107, 128, 150, 151, 152, 168, Shenoy, A V., 159, 177, 328, 329, 343 172, 177, 189, 219, 225, 229, 230, 238, Sheppard, N., 417, 422 266, 281, 284, 295, 311, 313 332 335 Sheptak, N., 517, 529 342-344 ' , , Shetty, R., 550, 556 Taylor, C. A, 522, 530 Shih, C. K, 407, 409 Tee, T. T., 223, 224, 230, 285, 296 Shirodkar, P., 249, 262, 267 Teyssie, Ph., 405, 409, 577, 600 Shirota, T., 393, 408 Thann, R c., 539, 555 Shroff, R N., 249, 267 Thomas, A, 286, 296 Shumsky, V. F., 172, 177 Thoone, J. H., 60, 66, 100 Shurcliff, W. A, 346, 348-350 352 363 T?rasher, KG., 411, 413, 417, 421 Sieglaff, C. L., 414, 421 " Tlemersma-Thoone, G., 174, 175, 177 S!mmons, J. M., 168, 177 Tiu, c., 317, 342 SltZ, C. E., 414, 422 Tob?lsky, A V., 99, 102, 188, 229,410, 421 Siva shinsky, N., 227, 230, 290, 291, 296 Tokl, S., 290, 296 Skolnick, J., 86, 101 Tokita, N., 130, 139, 151 Sm!th, F. P., 205, 229, 511, 529 Tong, P. P., 205, 229,511, 529 Smith, R. G., 78, 101,372, 389 Tonogai, S., 414, 422 Somer, K, 362, 364 Tordella, J. P., 362, 364 Soong, D. S., 227, 230, 290 291 294 296 297 ' , , , Torza, S., 403, 404, 409 Torzecki, J., 410, 421 Sorta, E., 290, 296 Tremblay, B., 318, 319, 340, 342 Soskey, P. R, 137, 152, 262 267 283 295 Sosulin, K N., 561, 565 ' , , Troup, G. J., 346, 363 Tsai, AT., 227, 230, 290, 291, 294, 296, 297 Souffie, RD., 539, 555 Tsang, W. K W., 204-206, 225, 229, 230, Sp~ncer, R S., 339, 344 511, 529 Spiers, R. P., 283, 295 Tschoegl, N. W., 43, 100 Spruiell, J. E., 523, 530 Tsunashima, Y., 615, 616 Srinivasan, R, 313, 342 Tuminello, W. H., 68, 69, 100, 574, 600 Starita, J. M., 415, 419, 422 561 564 565 566 ' , , , Tuna, N. Y., 313-315, 342 Tung, C. Y. M., 420, 423 Starr, F. 294, 297 c., Tungare, A V., 411, 419, 421 Stastna, J., 176, 178 Turner, S., 401, 409 Stephenson, S. E., 141-144, 152, 193, 229, 262, 264, 265, 267, 268 Stevens, M. J., 441, 457-459, 477, 489 Ulmer, AS., 498, 508 Stevenson, J. F., 265, 268, 335, 344 Umbach, H., 498, 508 Steward, E. L., 536, 555 Unsworth, J., 416, 422 Stockmayer, W. H., 615, 616 Ushida, Y., 552, 556 Stoehrer, B., 565, 566 Utracki, L. A, 174, 177, 334, 343, 407, 409 Stratton, R A, 163-165, 177,203,204,229 434, 440, 584, 600 ' 648 AUTHOR INDEX Valamonte, D., 471, 489 Williams, G., 219, 229, 493, 507 Valesano, V. A., 143, 145, 152 Williams, G. E., 438, 440 Valles, E. M., 280, 295 Williams, H. L., 416, 422 Van Aken, J. A, 262, 267 Williams, J. G., 495, 507 van de Hulst, H. c., 349, 363 Williams, L. c., 78, 101, 372, 406, 389 van der Veen, A, 565, 566 Williams, M. c., 290, 291, 296 Van Der Vegt, A K, 405, 409 Wilson, G. F., 510, 529 Van Krevelen, D. W., 358, 364, 370, 383, Wilson, N. R., 517, 529 384, 389, 619, 621 Winter, H. H., 72, 101, 137, 152, 181, 228, Van Dene, H. J., 405, 409 247,249,262, 267, 283, 295,406,409,420, Van Vijngaarden, H., 495, 507 423, 510, 529,549, 551, 556 Vergnes, B., 595, 600 Wissbrun, K F., 55, 100, 175, 178,247, 267, Villemaire, J. P., 329, 343, 499, 508 359,360,364,373,376,378,379,384,389, Vincent, M., 499, 508 411,421,426,432-437,439,440,518,521, Vinogradov, G. V., 166, 169, 172, 177,222, 523,525,529,530,544,555,570,578,579, 230, 332, 335, 338, 343, 344, 397, 409 590-593,598, 599, 600, 609, 612 Viola, G. G., 437, 440 Wong, C. P., 208, 209, 229 Vlachopoulos, J., 314, 342, 500, 508, 538, Worm, AT., 539, 555 555 Wortberg, J., 538, 555 Vlcek, J., 538, 555 Worth, R. A, 510, 529 Volgstadt, F. R., 414, 421 Worthoff, R. H., 304, 326, 341 Vrentas, C. M., 182, 187, 190, 192, 193, 228 Wrasidlo, W. J., 417, 422 Vu, T. K P., 285, 296 Wu, R., 537, 555 Wu, S., 68, 69, 100, 404, 405, 409,573, 599 Wagner, H. L., 376, 378, 384, 389, 614, 616 Wu, W. C. L., 516, 529 Wagner, M. H., 54,100,126,130,134, 138-144,151, 152, 193, 198,209,213,215, Yaloff, S. A, 417, 422 229, 240, 249, 251, 256, 257, 266-268 Yamada, N., 188, 229 Walters, K, 271, 294, 283, 284, 295, 303, Yamamoto, F., 438, 440 309,313,341, 342 Yamane, H., 174, 177, 181, 228, 341, 344, Wang, K K, 331, 343,505, 508 550, 556, 579, 600 Warashina, Y., 223, 230, 285, 296 Yamasaki, H., 415, 422 Watanabe, K, 411, 421 Yandrasits, M., 417, 422 Watanabe, R, 359, 364 Yang, B., 335, 344 Webb, P. c., 515, 526, 529 Yang, M.-C., 261, 267 Weeks, J. c., 515, 526, 529 Yang, T., 285, 296 Weese, J., 72, 101 Yang, W. P., 411, 413, 415, 416, 421 Weissberg, H. L., 321, 343 Yanovsky, Yu., 222, 230 Weissert, F. c., 395, 397, 398, 400, 408 Yap, C. Y., 416, 422 Wereta, Jr., A, 411, 413, 417, 421 Yaris, R., 86, 101,349,363 Werner, H., 486, 487, 490 Yasuda, KY., 163, 177 Wesselling, P., 495, 507 Yeh, P., 349, 363 Westover, R F., 385, 389 Yen, H.-C., 225, 230 White, J. L., 130, 139, 151, 172, 174, 177, Yokoi, H., 497,508 181, 228, 258, 259, 267, 290, 296, 318, Yoo, H. J., 397, 400, 409 334-336,338,339,341,342-344,359,364, Yottsutsuji, A, 503, 508 386,389,390,395,397,398,400,405,408, Yu, J. S., 538, 555 409, 497, 508, 523, 530, 547, 550, 555, 556, 579, 600 White, Jr., R P., 415, 422 Zachman, H. G., 498, 508 White, S. A, 318, 324, 342, 362, 364 Zahorchak, A c., 578, 579, 600 Whorlow, R W., 303, 341 Zapas, L. J., 127, 136, 140, 152, 182, 219, Wiest, J. M., 360, 361, 364 228, 229, 282, 295 Wiff, D. R., 55, 100 Zeichner, G. R, 68, 100, 561, 565 Wilcoxon, J. P., 420, 423 Zimm, B. H., 75, 101,614,616 Willey, S. K, 498, 508 Zukas, W. X., 416, 422 Williams, C. E., 577, 600 Ziille, B., 227, 230, 290, 291, 293, 296 Subject Index Acceleration, 40-41 Biaxial start-up flow, 261 Acetal copolymer, 610 Biaxial stress growth coefficient, 261 Activation energy for flow, 383, 584 Bingham plastic, 18, 398 effect of branching on, 388 Birefrigence, 349 Adhesives, 411 applications of, 362 Aerodynamic forces on a film bubble, 541 form, 350 Aerospace industry, 411 measurement of, 352-357 Air ring related to molecular orientation, 360 design of, 546-547 related to stress, 358 role of in film blowing, 531 use of to observe entrance effects, 319 Alignment, of molecules, 150-151 BKZ equation, 127, 129, 136, 149 American Society for Testing and Materials separable form of, 128, 266 (see ASTM) Blends Amplitude ratio, for oscillatory shear, 60 compliance of, 372 Anisotropy, 20 miscibility of, 406 optical, 349 Blends, immiscible Annular flow, as viscometric flow, 156 drop breakup in, 403 Antithixotropy, 17 drop deformation in, 403 Apparent extensional viscosity estimation of Tg for, 620 from converging flow test, 249 morphology of, 405 from melt spinning test, 248 role of drop elasticity, 405 Area swell (see Parison swell) role of Weber number in, 403 Arrhenius equation, 89, 383 size of disperse phase, 405 Aspect ratio, of filler, 393 viscosity of, 401, 402 ASTM, (American Society for Testing and Block copolymers, 575 Materials), 327, 328, 414, 415, SOl, 503, as immiscible blends, 402 596,614 rheological properties of, 407 Average molecular weights, 608-610 Blow molding (see a.lso Number average; Weight average; description of process, 509 Z-average; Viscosity average) flow in the die, 510 of engineering resins, 522 Bagley end correction, 320-322, 325 resin evaluation for 524, 525 effect of filler on, 400 resin quality control for, 528 Bannatek, 560 (see also Injection blow molding; Parison Barrier screws, 536 swell; Parison sag; Pleating; Parison Bernoulli's equation, 544, 545 formation; Parison inflation; Stretch Biaxial extension, 234 blow molding) experimental methods for, 262 Blow up ratio (BUR), 531, 534 role of in film blowing, 543 Blowability, 549 techniques for generating, 261 Blown film (see Film blowing) velocity distribution for, 260 Body force, 4, 5, 40 Biaxial extensional viscosity, 261 Bohlin Reologi, 622 649 650 SUBJECT INDEX Boltzmann superposition principle, 44-47, (see also Capillary flow; Entrance effects; 64,72-73, 104, 108, 112, 191, 193,208 Entrance pressure drop; Bagley end and oscillatory shear, 60 correction) and time-temperature superposition, 88 Carreau equation for viscosity, 162 applied to creep compliance, 58 Carri-Med, 622 applied to recoil, 58 Carter Baker Enterprises, 623 applied to step tensile strain, 46 Casson equation, 399 generalization of for nonlinear Cauchy strain tensor, 111 viscoelasticity, 115 Cauchy tensor, 109, 112 limitations of, 106 components for simple extension, 110 Box function for relaxation spectrum, 99 components for simple shear, 110 Brabender Instruments (C. W. Brabender definition of, 605 Instruments), 622 (see also Cauchy strain tensor) Branching, long-chain, 51, 576, 585 Cauchy's equation, 37, 39, 40 and MWD, 386 Ceast U. S. A., 623 effect of on activation energy, 388 Cessation of steady shear effect of on linear behavior, 99 for rubberlike liquid, 121 effect of on steady state compliance, 388 linear response, 73-74 effect of on time-temperature nonlinear behavior, 199 superposition, 92 Chain scission, 607 effect of on viscosity, 386-388 Channel flow Brinkman number, 467 converging, 329 Brownian motion, 19, 20, 42, 48-50, 75, 85, entrance effects, 317 357 irregular cross sections, 317 and entanglements, 142 use of lubrication approximation, 329 Bubble flow (see Film blowing) Weissenberg number for, 331 Bubble stability, 535, 550 (see also Converging flow) related to extension thickening, 237 Characterization of polymers, 573 Bueche-Ferry law, 79, 83 Chemorheology, 410 Bueche model (see Rouse model) Chromatography, 410 Bueche-Harding equation, 162, 378 Coating (see sheet extrusion) Coextrusion film blowing, 535 Capillary flow Coil-stretch transition, 360 apparent shear rate, definition, 300 Cole-Cole plot, 90 apparent viscosity in, 301 Collapsing ladder, 541 entrance pressure drop for, 322 Complex compliance, 69 of power law fluid, 301 (see also Storage and loss compliances) Rabinowitch correction, 303 Complex modulus, 63 Schuiimmer approximation, 304 strain amplitude-dependent, 219 shear rate, Newtonian fluid, 299 (see also Storage and loss moduli) shear stress distribution, 298 Complex viscosity, 64 slip velocity, 306 Compliance, of polymers, (see Creep wall shear rate, Newtonian fluid, 300 compliance; Steady state compliance) wall shear rate, power law fluid, 302 Compliance, of rheometers, 270, 271 wall shear stress, 299, 321-322 correction for, 273 wall slip in, 305 elimination of, 273 (see also Capillary rheometers; Channel Compounding, use of on-line rheometers for, flow; Entrance effects) 565 Capillary rheometers, 324 Compressibility, role of in extrudate apparent shear rate in, 326 distortion, 338 effect of pressure on, 327 Compression, 5 effect of wall slip, 327 Concentric cylinder rheometer, 285 for on-line use, 558 Condensation polymers, 576, 579 types of, 327 Cone and plate flow, 277 viscous heating in, 326 as a viscometric flow, 157 wall shear rate in, 325 (see also Rheometers) wall shear stress in, 325 Conformation, of molecule, 365 SUBJECT INDEX 651 Conservation of mass, 38 Use of supported samples to study, 416 Constitutive equations, 2, 37, 52, 107, 108 viscosity during, 418-419 Doi-Edwards, 148 Crystallinity, 595 for Newtonian fluid, 36 Crystallization, 574-576 nonlinear, 107 and glass transition, 619 (see also Continuum models) Cuff effect (see Parison swell) Constrained recoil (see Recoil) Cure time, 420 Continuity equation, 38, 39 Curing reactions (see Cross-linking reactions) Continuum, 38, 39 Curtaining (see Pleating) Continuum mechanics, 38, 107 Curtiss-Bird Model, 85 Continuum models 107, 127 Custom Scientific Instruments, 215, 623 Contour length fluctuations, 85 Contour length relaxation, 147, 148 Convected Maxwell Model, 117 Converging flow, 236 Damping function, 131 Deborah number for, 330 and Cox-Merz rules, 174 lubricated, 249 and Gleissle mirror relationships, 175 pressure drop in, 330 and irreversibility, 142 tensile extension in, 323 and ultimate recoil following steady simple use of lubrication approximation for, 329 shear, 216 use of to measure apparent extensional comments on use of, 144 viscosity, 249, 323 definition of, 130 use of for quality control, 598 dependence on temperature, 136 (see also Entrance effects; Entrance determination of, 131 pressure drop) equations for, 134 Copolymers, 576 exponential, 135 estimation of Tg for, 620 for extensional flows, 133, 138, 239-240 Costech Associates, 623 for multiaxial flow, 266 Couette flow, as viscometric flow, 156 for polystyrene, 184 Coupling agents, effect of on suspension yield for relaxation strengths, 190 stress, 397 from Doi-Edwards theory, 149 Cox-Merz Rules, 173-175,584 in terms of entanglements, 141 for block copolymers, 407 prediction of tensile and shear recoil using, Creep, 14,55 143 methods of measuring, 286 relationship to shear stress growth (see also Extensional creep) coefficient, 196, 198 Creep compliance, 55, 56 shear, 132 for typical polymer, 58, 59 shear, equations for, 136 nonlinear, 207 shear, for various polymers, 137 time-temperature superposition, 209 universal, 139 of polystyrene, 94 Damping functional, 142 Creep recovery (see Recoil) Data acquisition systems, 572 Creeping flow, 40, 41 Daventest Limited, 624 Creepmeter Deborah number rotational, 286 for converging flow, 330. sliding plate, 291 for die flow, 332 tensile, 242-243 for oscillatory shear, 220 Cross equation for viscosity, 162, 378 Deformation, 1, 6, 8, 605 Cross-linking reactions, 410, 411, 607 stretching, 232 gel point of, 420 large, rapid, 103 of LLDPE for film blowing, 552 Deformation gradient tensor, 35 models for, 419 Degradation, 607 rate of, 419 controlled, of polypropylene, 565 relaxation modulus during, 420 correction of rheological data for, 580-581 storage and loss moduli during, 420 due to moisture, 579 use of dielectric analysis to monitor, 417 in film blowing, 536 use of rheometers to monitor, 413-414 use of rheology to monitor, 410 652 SUBJECT INDEX Devolatilization, in extruders, 445, 484 Draw resonance Diameter swelHsee Parison swell) in sheet extrusion, 554 Dichroism, 350 use of for quality control, 598 Die design, 332 Drawability, related to extension thickening, Die flow 237 in blow molding process, 510 Ductile failure, in simple extension, 242 in coat hanger die, 595 Dupont Company, 624 Die swell (see Extrudate swell) Dynamic mechanical analysis, used to study Dielectric analysis, 417 curing reactions, 416 Differential scanning calorimetry, 410, 574, Dynamic spring analysis, 416 618 Dynamic viscosity, 63 Differential thermal analysis, 618 Disclinations, 430 Discrete spectrum, 53 Eccentric rotating disk rheometer (see Disengagement time, 83, 147 Rheometers) Displacement functions Edge bead in sheet extrusion, 552 definition of, 601 Edge effects (see End and edge effects) for simple extension,,606 Einstein summation convention, 25 for simple shear, 601 Einstein's equation, 390 Displacement gradient tensor Elastic energy potential, 127 definition of, 602 Elastic energy storage, 17 for simple .extension, 606 Elasticity, 10 for simple shear, 605 Elastomer, 2 Displacement vector, 32 Electric vector, 347 Dissipation Electronics industry, 411, 414 and structural dependency, 16 End and Edge Effects effect of on temperature distribution in in cone-plate rheometers, 280 rheometers, 274 in drag flow rheometers, 275 in dash pot, 15 in rheometers, 158 in extruders, 453, 461 End effects in capillary flow (see Entrance in oscillatory shear, 63 effects; Entrance pressure drop; Exit Diverging flow, use of to generate biaxial pressure; Bagley end correction) e~tension, 262 End-to-end distance, of molecule, 19-20 Doi-Edwards Constitutive Equation, 148 Energy dissipation (see Dissipation) damping function from, 149 Entanglement coupling, 79 prediction of viscosity by, 149 nature of, 80 Doi-Edwards theory, 82-85, 183, 184 Entanglement density, 141 damping function for shear, 137 related to viscosity, 167 double step strain, 192 Entanglements, 21, 49-50, 80, 83, 358 nonlinear viscoelasticity, 146-149 effect of on viscosity, 164 prediction for double step strain, 193 evidence for existence of, 79 strain dependent relaxation modulus Entrance effects prediction, 147 for filled melts, 400 stress growth coefficients for, 199 in channel flow, 317 Doolittle equation, 382 observations of in capillaries and slits, 318 Double-step reversing strain test related to extrudate distortion, 339, 341 effect of irreversibility assumption, 143 vortex flow pattern, 318 Double step strain, 191 Entrance flow (see Entrance effects; prediction of Wagner's equation, 192 Entrance pressure drop) Drag flow, 156 Entrance pressure drop, 319 combined with pressure flow, 157 and viscoelasticity, 323 sliding surface, 269 and extensional flow, 323 Drag flow rheometers (see Rheometers) and extensional viscosity, 323 Draping (see Pleating) and recoverable shear, 323 Draw down (see Parison sag) for Newtonian fluid, 320-321 Draw down ratio (DDR), 531, 534 in melt indexers, 328 SUBJECT INDEX 653 related to orifice pressure drop, 322 role of fluorocarbons, 539 (see also Bagley end correction) role of slip, 340 Epoxy curing, 415 spurt effect, 338 Equation of state (see Constitutive equation) types of, 336 Equibiaxial extension (see Biaxial extension) Extrudate drawing Equilibration time, 83, 146 analysis of, 247 Equilibrium modulus of elastomer, 49 to generate uniaxial extension, 246 Exit pressure use of to measure apparent extensional determination of N\ from, 313-314 viscosity, 247 measurement of, 315, 560 isothermal, 248 Exponential shear, 225 (see also Melt spinning) compared with planar extension, 151, 265 Extrudate swell Exponential viscosity, 226 definition of, 332 Extensiometers (see Rheometers, dependence on temperature, 335 extensional) dependence on MWD, 335 Extension thickening behavior, 254 dependence on time and temperature, 334 definition of, 235 effect of filler on, 400 effect of branching on, 260 effect of LID on, 333 ofHDPE,259 for capillary, 332 Extension thinning behavior for Newtonian fluid, 332 definition of, 235 for noncircular dies, 335 Extensional creep, 58 in film blowing, 540, 544 (see also Tensile creep compliance) methods for measurement of, 334 Extensional flow, 231, 236 . of block copolymer, 408 at entrance to capillary, 323-324 of HDPE, 333, 334, 336 coordinate system for, 233 of polypropylene, 334 definition of, 232 prediction of, 335 genera], 265 related to N\, 335 multiaxial, 265 theoretical analyses of, 335 rate of deformation tensor for, 265 ultimate value of, 334 role of in blow molding, 527 use of for quality control, 597 Extensional recoil (see Tensile recoil) (see also Parison swell) Extensional rheometers (see Rheometers) Extruders, 441 Extensional stress growth coefficient, role of adiabatic flow in, 469 in film blowing, 543 analysis of, 441 Extensional viscosity (see Tensile viscosity) characteristic curves for, 463-466 Extinction angle, 360 coupled with dies, 454 Extinction coefficient, 349 devolatilization in, 484 Extinction of light, 354 drag flow and pressure flow in, 451 Extra stress, 31 effects of simplifying assumptions on, 459 in Newtonion fluid in simple shear, 36 efficiency of, 453 Extrudate distortion feed zone in, 470 critical stress for, 539 functions of, 442 effect of die material, 340 leakage flow in, 453, 460, 461 effect of resin additives, 340 melt conveying zone, 446 factors it depends on, 337 melting zone of, 472 gross melt fracture, 339, 341 mixing in, 480-484 in blow molding process, 511, 527 modelling of, 459-469 in film blowing, 534, 538, 539 non-isothermal flow in, 467 of HDPE, 338, 339 operating diagrams for, 455-459 of LDPE, 341 plasticating, 444 of LLDPE, 339 power consumption in, 461 of PDMS, 340 role of Brinkman number in, 467 of polystyrene, 339 scale-up of, 476-480, 587 oscillatory flow, 338 screw zones of, 444 related to entrance flow, 339 single-screw, simple model of, 446-454 654 SUBJECT INDEX Extruders (cont'd.) use of blends for, 552 solids conveying in, 470 viscous stresses, 543 twin screw, 485-489 (see also Draw down ratio; Air ring; types of, 443 Bubble stability) use of power law to model, 468 Finger strain tensor, 111 velocity distribution in, 449-452 Finger tensor, 109, 112, 130 viscous dissipation in, 453, 461 components for simple extension, 110 Extrusion (see Extruders) components for simple shear, 110 Extrusion blow molding (see Blow molding) definition of, 113, 605 Extrusion casting (see Sheet extrusion) for simple shear, 113 Extrusion coating (see Sheet extrusion) (see also Finger strain tensor) Extrusion plastometer (see Melt index) Finite linear viscoelasticity, 108, 115 First normal stress coefficient definition of, 155 Fading memory, 22, 47 dependence on relaxation spectrum, 171 Fiber-filled melts, flow of, 399 effect of molecular weight on, 170 Filled melts effect of shear rate on, 173 elasticity of, 400 for rubberlike liquid, 120 entrance effects for, 400 related to storage and loss moduli, 176 extrudate swell of, 400 related to viscosity, 176 first normal stress difference of, 400 relationship to steady state compliance, viscosity of, 393 121 (see also Viscosity) relationship to storage modulus, 121 Film blowing, 247 relationship to relaxation modulus, 120 aerodynamic forces, 544, 546 temperature dependence of, 171 air ring design, 546-547, 552 time-temperature superposition of, 173 bubble shape, 542, 543, 547-548 First normal stress decay coefficient, 200 bubble stability, 548, 550 First normal stress decay function, 200 bubble temperature, 547 First normal stress difference, 170 cooling air flow, 545 for a purely elastic linear rubber, 123 cooling rate, 547 for lart~e amplitude oscillatory shear, 224 description of process, 531 of filled melt, 400 die flow, 538 of LDPE, 316 drawability, 549 of liquid crystal polymers, 437 extrudate distortion, 538, 539 of rubberlike liquid, 120 extrudate swell, 536 related to exit pressure, 313-314 flow in extruder and die, 536 related to shear stress, 172 flow in the bubble, 540 relationship with steady state compliance forces acting on bubble, 541-543 at low .y, 170 gauge variations, 551 sources of error in measurement of, 282 linear and branched polyethylenes, 552 use of cone-plate rheometer to measure, maximum draw ratio, 549 281 molecular orientation, 534, 544 use of hole pressure to determine, 312 objectives of process, 533 use of slit rheometer to determine, 309 of polyethylene, 551 (see also First normal stress coefficient; process optimization, 590 Normal stress differences) production problems, 534 First normal stress growth function, 195 resins used for, 536 for rubberlike liquid, 120 role of branching, 549 Flow in channels (see Channel flow) role of MWD, 549, 551 Flow rate (see Melt index) role of rheological properties, 533 Fluid mechanics, 37 selection of resins for, 590 Fountain effect, 362, 495 sharkskin, 533, 534 Fourier series, for large amplitude oscillatory special extruders for, 536 shear, 223 strain history, 544 Free volume, 382-383 strain rates in bubble, 543, 544 role of in glass transition, 618-619 SUBJECT INDEX 655 Friction coefficient, of molecule, 75-76 related to N!, 310-311 Frost line, in film blowing, 531, 544 use of to determine N!, 312, 560 Hooke's Law for extension, 10 Gate flow (see Injection molding) for simple shear, 11 Gaussian normal error curve, 611 Hydrodynamic interaction, in polymer Gel point solutions, 75 definition of, 420 Hydroxypropylcellulose, 425 determination of, 420 Gel state, 411 Gel time, 420 Immiscible blends, rheological properties of, Gelation (see Cross-linking reactions) 406 Generalized Maxwell model, 52-53 In-line rheometers, 562 and Lodge's network theory, 117 compared with on-line rheometers, 558 and steady state compliance, 56 (see also On-line rheometers) and time-temperature superposition, 87 Incompressibility of polymers, 29, 30, 38,113 determination of parameters for, 71 Independent alignment assumption, 148, 149, storage and loss moduli for, 64 193 Glass transition temperature, 19,50-51,382, Infinitesimal strain tensor (see Strain tensor) Infrared spectroscopy, 411 574 Inherent melt viscosity, 614 definition of, 617 effect of molecular structure on, 619 Inherent viscosity, 614 effect of plasticizers on, 619 role of in injection blow molding, 523 measurement of, 617 Injection blow molding of blends, 620 description of process, 509 of copolymers, 620 of PET, 523 related to melting point, 620 stretch blow molding, 523 Glassy behavior Injection molding of polymers, 50-51, 66 controlled rheology resins for, 506 of elastomer, 48 description of process, 491 of melt, 49 drooling, 499 Gleissle Mirror relations, evaluation of, 175 elongational flow, 498 G6ttfert, 245, 559, 624 evaluation of resins for, 500 Rheotens, 247 flash, 499 flow in runners, 492 fountain effect, 495 UBI/Haake, 625 frozen wall layer, 495 HDPE (High density polyethylene) gate flow, 494 damping function, extension, 139 microstructure formation, 500 entrance flow of, 318 modelling of, 499 entrance pressure drop of, 320 mold cavity flow, 494 extensional flow of, 258 mold design for, 499 extrudate distortion of, 338, 339 mold ability tests, 502 extrudate swell of, 333, 334, 336 molecular orientation, 497 film blowing of, 536, 548, 549, 551 multiple live-feed, 498 MWD of, 611 of liquid crystal polymers, 438 parison swell of, 514 of thin parts, 499, 506 tensile viscosity of, 258 optimal operating conditions, 498 use of for film blowing, 535 process objectives, 491 Helical flow, as viscometric flow, 157 residual stresses, 497 Hencky strain, 8, 35, 606 resin evaluation for, 505-506 Hencky strain rate, 9, 237 rheological models for, 587 High density polyethylene (see HDPE) role of constitutive equation, 500 Hole pressure role of melt index, 501 definition of, 310 role of MWD, 500-501 of Newtonian fluid, 310 role of rheological properties, 502 on-line measurement of, 560 selection of resins for, 506 656 SUBJECT INDEX Injection molding (cont'd.) edge fracture in, 281 shear rate in mold, 495 entrance flow of, 318 short shots, 494 exit pressure of, 315 use of birefringence to measure stresses, exponential shear, 227 362 extensional flow properties, 253 use of on-line rheometers for, 565 extrudate distortion of, 341 viscosity models for, 499 film blowing of, 536, 539, 540, 547, 549, (see also Runners; Weld line; Jetting; 551 Spiral mold test) film resins, 181 Instron, 625 first normal stress difference, 315 Instrument compliance (see Compliance, of melt strength of, 248 rheometers) nonlinear creep, 213 Interfacial tension nonlinear relaxation modulus of, 133, 188 effect of on blend viscosity, 402 planar extension of, 265 Internal bubble cooling, 534 recoil during tensile start-up flow, 143 Interrupted shear, 203 recoil following steady shear, 214, 216 Intrinsic viscosity, 615 recoil function of, 212 Invariants (see Scalar invariants) relaxation modulus of, 99 Inverse of a tensor, 603n sheet extrusion of, 553 Irreversibility assumption, 144, 194 simple extension of, 244 and Wagner's equation, 192 start-up of steady simple shear, 134 effect of on tensile and shear recoil steady state compliance of, 254 predictions, 143 stress growth coefficient for, 198 in Wagner's equation, 142 stress growth functions, 214 Isotropic stress, 29, 30 tensile creep of, 243 IUPAC A (LDPE), 249-251, 257 tensile recoil of, 249 IUPAC C (LDPE), 540 tensile stress growth function of, 126 IUPAC X (LDPE), 227 tensile stress growth coefficient of, 250 tensile viscosity of, 254, 255 Jetting, in injection molding time-temperature superposition applied causes of, 497 to, 188 elimination of, 497 use of for film blowing, 535 Jones matrix, 352, 354 viscosity of at several temperatures, 160 Leslie-Erickson constitutive equation, 427-428 Karl Frank, 624 Leslie-Erickson theory for liquid crystals, Kayeness, 625 426-427 Kelvin body (see Voigt body) Light absorption, 347-349 Killion, 625 Light scattering, 347-349, 360 Kinematics, 45, 103, 104, 150, 180, 232, 238, Light transmission, 354 260 Linear low density polyethylene (see change of with time, 151 LLDPE) - Knit line (see Weld line) Linear polymers, simple extension of, 258 Kronecker delta, 111 Linear strain, for simple extension, 7, 35 Liquid crystal polymers (LCPs) Large amplitude oscillatory shear (see birefringence of, 359 Oscillatory shear) commercially available, 425 Laser Doppler velocimetry, 346 domain model of, 432-433 Laser speckle interferometry, 346 extrudate swell of, 438 Layflat width, 532 first normal stress difference of, 437 LDPE (Low density polyethylene) lyotropic, 424 cessation of steady simple shear, 203 molding of, 438 interrupted shear, 204 nematic, 424 damping function, shear, 135, 198 processing of, 437 damping function, extension, 138, 139, 257 shape of molecule, 425 SUBJECTr INDEX 657 thermotropic, 424 Melt drawing (see Extrudate' drawing) validity of stress-optical relation for, 360 Melt fracture viscosity of, 431-437 in oscillatory shear, 222 Liquid crystals (see also Extrudate distortion; Wall slip) continuum theory of, 426 Melt I (LDPE), 133-135, 138,.160, 188, 190, disclinations in, 430 198, 203, 213, 214, 216,249 viscosity of, from L-E theory, 428 Melt index, 159; 327 viscosity of, measured, 429-430 effect of evolved gas; 577 LLDPE (Linear low density polyethylene) measurement of, 328 double step strain, 143 on-line measurement of, 559 draw resonance in, 554 role of in blow molding, 528 extrudate distortion of, 339 role of in extrusion, 462 film blowing of, 536, 538, 539, 547, 549, role of in injection molding, 501, 506, 507 551, 552 use of for quality control,597 sheet extrusion of, 554 Melt index elasticity test; 215' slip velocity of, 306, 307 Melt spinning use of for film blowing, 535 use of birefringence in, 362 Lodge-Meissner relationship, 133, 148, 190 role of viscosity in, 586 Lodge Stressmeter, 560 use of for quality control, 598 Lodge's network theory, 117 (see also Extrudate drawing) use of to predict parison sag, 520 Melt strength, 247, 248 Log normal MWD, 610 related to parison swell, 528 Long-chain branching (see Branching) role of in extrusion casting; 554 Loss modulus (see Storage and loss moduli) role of in film blowing, 549 Low density polyethylene (see LDPE) use of for quality. control; 598 Lubricated squeezing flow, to generate Memory, of viscoelastic material, 22 biaxial extension, 262 Melting point, 574 Lubricating oil, 2 related to Tq, 620 Lubrication approximation, 329 Memory function for converging flow, 331 from Lodge's network theory, 117 in simulation of melt processing, 330 nonlinear, 131, 190, 201 relationship with relaxation time, 115 relationship with spectrum function, 116 Manufacturers of rheometers, 622 separability of, 130, 132 Mark-Houwink-Sakurada equation, 615 strain dependent, 128, 129 Maron-Pierce equation, 395, 615 Mesogens, 424 Master curve, 87 Mesophase, 424 Master Unit Die Products, 504 Metravib, 245, 626 Material constant, 3, 11 Metrilec Sari, 329, 626 Material functions, 3, 74 Microdielectrometer, 417 for steady simple shear, 155 Mitech Corporation, 623 of nonlinear viscoelasticity, 106 Mixing, in extruders, 480-484 Maximum packing fraction, of suspension, Modified Rouse theory (see Rouse model) 393 Modulus of rigidity, 11 Maxwell element, 14,43,52 (see also Shear modulus) (see also Maxwell model) Moisture, absorption of by polymers, 579 Maxwell model, 52 Moldability tests, 502 exponential shear predicted by, 226 Molecular dynamics, 107, 108 (see also Generalized Maxwell model) Molecular models for nonlinear Maxwell model parameters viscoelasticity, 146 determination of, 70 Molecular motions, 367 strain-dependent, 188 Molecular orientation, 214 MECA Creep Rheometer, 286 in blown film, 533 Mechanical loss angle, 60 Molecular stretching in simple extension, 238 for several polymeric liquids, 61 Molecular structure, 366 658 SUBJECT INDEX Molecular theories Normal stress, 6 for entangled melts, 79 Normal stress differences, 31, 154, 155 overview of, 365 for Newtonion fluid in simple shear, 37 prediction of linear behavior by, 74 for Newtonion fluids, 30 Molecular weight for rubberlike liquid, 118 effect of on viscosity, 368 second, prediction of by Doi-Edwards average, between entanglements, 51 theory, 190 Molecular weight distribution (MWD) Normal stress relaxation functions effect of on steady state compliance, 373 for rubberlike liquid, 119 effect of on viscosity curve, 376-381 Nuclear magnetic resonance (NMR), 619 description of, 607 Number average molecular weight, 609 determination of from rheological Nylon measurements, 574, 585 MWD of, 610 effect of on extensional flow properties, 259 effect of on tensile viscosity, 254 On-line rheometers log normal, 373, 378, 610 applications of, 557, 563, 564 measurement of, 611 capillary type, 558 most probable, 610 compared to in-line rheometers, 558 Monohole flow test, 414 types of, 558 Monsanto Instruments, 626 Operating diagrams Morphology, of a blend, 402 for blow molding, 592 Most probable MWD, 610 for extruders, 455-459 MTS, 294, 626 Optical axes, 349 Mueller matrix, 352, 354 Orientation, of molecule, 20, 227 Multiphase systems, 390 Orifice flow, pressure drop for, 322 Multistep strain, 191, 193 Oscillatory shear MWD (see Molecular weight distribution) effect of fluid inertia on, 276 large amplitude, 219, 222, 223 Nametre, 626 limits of linear behavior, 222 Navier-Stokes Equation, 40 of rubberlike liquid, 121 Neck-in, 554 small ampli tude, 60 Net stretching stress, 31 superposed on steady shear, 218 for biaxial extension, 260 use of to probe relaxation, 219 for step strain of linear material, 46 (see also net tensile stress) Net tensile stress, 43, 58, 238 p-azoxyanisole, 429 Newton's law of action and reaction, 27 p-hydroxybenzoic acid, 425 Newton's second law of motion, 37, 39 Parallel disk flow, as viscometric flow, 157 Newtonian fluid, 2, 3, 36, 41 Parallel disk rheometer, use of to study oscillatory shear of, 62 curing reaction, 417 as special case for power law, 161 Parison formation, 590 dilute suspension in, 390 of engineering resins, 522 entrance pressure drop of, 321 modelling of, 521 in simple shear, 36 role of rheological properties in, 525 normal stress differences, 30 Parison inflation pressure drop in orifice, 322 as extensional flow, 237 simple extension of, 37 modelling of, 522 slit flow of, 308 role of rheological properties in, 527 viscosity of, 11 Parison programming, 510, 512 viscosity of from tube flow data, 301 Parison sag, 512 Nip rolls, 531, 532, 541 as example of uniaxial extension, 236 No-slip assumption, 158 combined with swell, 520 Nonlinear viscoelastic behavior, 104-107, 179 of engineering resins, 522 classification of, 145 measurement of, 520 rheometers for study of, 270 related to operating conditions, 526 SUBJECT INDEX 659 related to resin properties, 526 Polybutadiene related to rheological properties, 519 normal stress differences for, 191 Parison swell, 512, 513 stress growth coefficients for, 199 combined with sag, 520 Polycarbonate, 574 cuff effects, 514 Polydimethylsiloxane (PDMS) definitions of, 513 cross-linking of, 420 effect of die design on, 515 damping function for, 140 effect of flow rate on, 515 entrance flow of, 319, 340 effect of MWD on, 515 extrudate distortion of, 340 effect of temperature on, 514 Polydispersity effect of die design on, 516, 517 effect of on relaxation spectrum, 98, 99 importance of, 514 effect of on steady state compliance, 86 isotropic, 515 (see also Molecular weight distribution) isotropic versus anisotropic, 516 Polyethylene, 95 measurement of, 517, 526 molecular structure of, 95 prediction of, 519 radius of gyration of, 21 related to capillary swell, 518, 527 time-temperature superposition applied related to stress ratio, 519 to, 92 relationship between different types, 513, volume of the sphere occupied by, 21 516 (see also HDPE; LDPE; LLDPE) time dependence of, 514 Polyethylene terephthalate (PET), 384, 575 PDMS (see Polydimethylsiloxane) absorption of water by, 579 PET (see Polyethylene terephthalate) blow molding of, 521, 523 Pillow mold (see Parison swell, measurement crystallization of, 619 of) glass transition of, 619 Pinch-off mold (see Parison swell, injection blow molding of, 523 measurement of) MWD of, 610 Pipkin diagram for oscillatory shear, 220, 222 Polyisobutylene, 219 Planar extension, 234 bidirectional start-up flow of, 199 as a strong flow, 265 cessation of steady simple shear, 201 comparison with exponential shear, 151 damping function for, 136 material functions of, 264 extensional flow of, 266 methods for studying, 264 Polyisobutylene solution of LDPE, 265 damping function for, 140 rate of deformation tensor for, 263 Polymer laboratories, 627 role of in film blowing, 543 Polymer molecules, why elastic, 19 velocity distribution for, 263 Polymer solutions, 19 Plasticity, 18 Polymerization reactions, 576 Plastication, in extruders, 442 Polymethyl methacrylate (PMMA), 162, 370 Plasticizers, 619 extensional flow of, 259 Plateau compliance, 58 Polypropylene effect of polydispersity on, 60 extrudate distortion of, 339 Plateau modulus, 50, 51 extrudate swell of, 334 in Doi-Edwards theory, 83 film blowing of, 547 related to steady state compliance, 372 MWD of, 561, 611 Plateau zone 50, 66 parison swell of, 514 Pleating, in blow molding, 521, 590 use of rheometer for, 561 PMMA (see polymethylmethacrylate) Polystyrene, 162, 190, 370, 598 Poiseuille Flow, as viscometric flow, 156 compliance of, 93 (see also Capillary flow) damping function for, 135, 137, 140 Polarizability, 358 extensional flow of, 259 Polarization diagrams, 348 extrudate distortion of, 339 Poly(n-octyl methacrylate), storage nonlinear creep compliance of, 208 compliance of, 90 nonlinear relaxation modulus of, 188 Poly-p-benzamide, 425 , normal stress differences for, 191 Poly-p-phenylene teraphthalamide, 425 relaxation modulus of, 183 660 SUBJECT INDEX Polystyrene (cont' d.) Reactive extrusion, use of on-line rheometers relaxation spectrum of, 98 for, 565 shear stress growth coefficient of, 196 Recoil start-up of steady simple shear, 195 during creep experiment, 57 storage and loss moduli of, 95-97 during start-up flow, 214 tensile creep, 243 methods of measuring, 286 time-temperature superposition of, 93 (see also Ultimate recoil; Tensile recoil) type II nonlinear behavior of, 196 Recoil function, 57 viscosity curves for, 163 for typical melts, 58-59 Polystyrene solution nonlinear, 211 damping function for, 140 Recoverable compliance (see Recoil Polyurethane function) cross-linking of, 420 Recoverable shear (see Ultimate recoil) Polyvinyl chloride (PVC), 434, 584 Recoverable shear strain (see Recoil) die flow of, 510 Recoverable strain, 257 Position vector, 32 at high strains, 253 Potential function of polystyrene, 259 of factorable BKZ model, 128 (see also Ultimate recoil; Ultimate tensile Power law for viscosity, 160-162, 166, 376 recoil; Tensile recoil) for liquid crystal polymers, 435 Reduced viscosity, 613 in capillary flow, 301 Reduction in shear rate test, 205 in injection molding, 493 Reentanglement time use of to model flow in mandrel die, 538 for interrupted shear, 204 use of to model die flow, 510 for reduction in shear rate, 205 use of to model extrusion, 464 Reference configurations, 112 Pressure, 29, 30, 39 Reference state, 7, 112 Pressure, effect of on viscosity, 385 Refractive index Principal strain axes, 233 as tensor quantity, 349, 357 Principal stresses, extensional flow, 238 definition of, 346 Principal stretching stress (see Net tensile imaginary, 349 stress) related to stress, 358 Process control Relative viscosity, 613 manipulated variables for, 564 of a suspension, 393 use of rheometers for, 564 Relaxation modulus, 44, 48, 51 Processability 180, 181, 584 Doi-Edwards prediction of, 83 Processing aids, use of in film blowing, 539 for Maxwell element, 52 Pseudoplasticity (see Shear thinning empirical equations for, 51 behavior) effect of polydispersity on, 51 PVC (see polyvinyl chloride) nonlinear, 132, 133, 183, 189 of cross-linking material, 420 of typical polymeric materials, 49 Quality control, 104, 159, 594-599 power-law form of, 54 use of solution viscosity for, 615 relationship to memory function, 116 selection of rheometer for, 596 relationship with storage and loss moduli, for blow molding resins, 528 64 separability of, 132, 184 Rabinowitch correction, for capillary flow, type I nonlinear behavior, 186 303, 325, 326 type II nonlinear behavior, 186, 188, 193 Radius of gyration, of branched molecule, Relaxation spectrum, 54 387 and Maxwell parameters, 72 Random walk model, 357 from experimental data, 55 Rate of deformation, for simple shear, 35 of polystyrene, 98 Rate of deformation tensor, 34 relationship to storage and loss moduli, 64 for extensional flows, 233 shear rate-dependent, 167 Reaction injection molding (RIM), 411 strain-dependent, 188 rheometers for study of, 416 truncation of at high shear rates, 167, 168 SUBJECT INDEX 661 Relaxation strengths Rheometries, 241, 245, 286, 560, 561, 564, and recoil, 216 627 from tensile viscosity, 257 Rheopexy, 17 shear-rate dependent, 202 Rheoplast, 329 Relaxation time Rheoprocessor, 561 and tensile stress growth function, 125 Rheovibron, 416 longest, 169 RIM (see Reaction injection molding) longest, from Doi-Edwards theory, 84 Rise time, in step strain, 181 longest, Rouse, 77, 147 Rod climbing (see Weissenberg effect) of a Maxwell element, 16 Rosand Precision Limited, 627 role of in oscillatory shear, 220 Rotary clamp, 243 effect of temperature on, 87 use of to measure melt strength, 247 from modified Rouse theory, 76 use of with HDPE, 258 of generalized Newtonion model, 71 use of to generate biaxial extension, 262 Reptation, 81, 83, 85, 86, 146 Rouse model, 146 Resin optimization, 588-595 for dilute solutions, 74 Resin selection, 585 and polydispersity, 78 Rest time, for interrupted shear, 203 Bueche modification of, 75 Retarded elasticity, 14 nonlinear effects, 146 Retraction time, 148, 184 prediction of linear viscoelastic properties, Reynolds number, 41 77 Rheo-optics, 345 Rouse theory, 371, 377 Rheological testing, 568-577 Rubber, 2, 3, 7, 13, 25 Rheology optical properties of, 350 aspects of, 2 polarization of, 350 definition of, 1 refractive index of, 350 Rheometer compliance (see Compliance, of Rubberlike liquid, 201 rheometers) and spike strain test, 193 Rheometers cessation of steady shear, 121 automation of, 572 comments on, 126 concentric cylinder, 285 deficiencies of, 126 cone-plate, 277 definition of, 115 advantages of, 278 exponential shear of, 228 basic equations for, 278 for any simple shear deformation, 115 edge effects in, 280-281 generalization of, 130 sources of error for, 279 in oscillatory shear, 121 controlled stress, 286 in simple extension, 123 drag flow, sources of, 270 in simple shear flows, 118 eccentric rotating disks, 284 in steady simple shear, 119 extensional, 242, 244, 245 in step shear strain, 119 industrial use of, 567 normal stress differences for, 118 manufacturers of, 622 recoil of, 122 parallel disk, 283, 286 ultimate recoil following steady simple sandwich, 291 shear, 216 selection of, 567-573 Runners sliding cylinder, 294 design of, 493 sliding film, 293 flow balancing in, 493 sliding plate, 287 flow in, 492 sliding plate: basic equations for, 288 heated, 492 design of, 291 (see also Injection molding) end and edge effects, 289 uses of, 290 sliding surface, 269 use of to measure N\, 282 Sag (see Parison sag) (see also Torque rheometers; Capillary Sandwich rheometers (see Rheometers) rheometers; Slit rheometers) Scalar invariants of the Finger tensor, 129 662 SUBJECT INDEX Scalar invariants of the Finger tensor (cant' d.) Shear stress growth function, 194 dependence of damping function on, 130, (see also Shear stress growth coefficient) 140 Shear stress transducer for simple extension, 131 use of in an extruder, 562 for simple shear, 131 use of in an in-line rheometer, 562 for simple extension, 114 Shear thickening behavior, of liquid crystal Scalar invariants of a vector, 113 polymers, 437 Schiimmerapproximation for wall shear rate Shear thinning behavior, 12 in capillary flow, 304, 326 effect of MWD on, 376 Screen pack, 531 Shear Wave Propagation, in drag flow Screening of resins, 573, 593 rheometers, 275 Second normal stress growth coefficient, 195 Sheet extrusion, 247, 552 Second normal stress difference, 154, edge bead, 553 and BKZ equation, 129 draw resonance, 554 prediction of by Doi-Edwards equation, neck-in, 554 149 role of melt strength, 554 (see also Normal stress differences) Sheet inflation Second Normal Stress Coefficient use of to generate planar extension, 264 definition of, 155 use of to study biaxial extension, 261 Second normal stress decay function and Shift factor, for time-temperature coefficient, 200 superposition, 88, 89, 94, 173 Secondary flow, in cone-plate rheometers, as a function of temperature, 93 279 for viscosity, 169 Seiscor, 313, 559, 560, 627 Shimadzu Scientific Instruments, 628 Seiscor/Han Rheometer, 313, 560 Simple extension, 7, 29, 31 Separability definition of, 234 of memory function, 128, 145 displacement functions for, 606 of relaxation modulus, 184 displacement gradient tensor for, 606 Sharkskin, 337-338 infinitesimal strain tensor for, 35 (see also Extrudate distortion) of Newtonion fluid, 37 Shear compliance, 11 shear rate tensor for, 35 Shear creep compliance (see Creep velocity distribution, 237 compliance) Simple shear, 6, 9, 28 Shear-free flows, 233 coordinate system for, 233 (see also Extensional flows) displacement function for, 601-602 Shear modification, 205 invariants of Finger tensor for, 113 effect of on extensional flow, 540 rate of deformation tensor for, 36 in blow molding process, 511 rheometers, 269 in film blowing, 540 stress components for, 29, 154 role of branching in, 388, 540 (see also Steady simple shear) Shear modulus, 11 Simulation of melt processes, 587 (see also Relaxation modulus) Sinusoidal shear (see Oscillatory shear) Shear rate, 10 Sliding cylinder flow, as viscometric flow, 157 in simple shear, 153 (see also Rheometers) Shear refining (see shear modification) Sliding plate rheometers (see Rheometers) Shear relaxation modulus (see Relaxation Slip flow (see Wall slip) modulus) Slit flow Shear strain, 9, 10 advantages of, 307 in simple shear, 153 as viscometric flow, 156 Shear stress, 6, 28 determination of wall shear rate in, 309 Shear stress decay coefficient, 73, 200, 201 of Newtonian fluid, 308 Shear stress decay function, 199 shear stress distribution in, 307 Shear stress growth coefficient, 72, 73, 195 use of to determine Nt, 309 as a function of shear strain, 112 wall shear stress in, 308 related to tensile stress growth coefficient, (see also Slit rheometers; Entrance effects; 231,239 Hole pressure; Exit pressure) SUBJECT INDEX 663 Slit rheometers calculation of, 220 for on-line use, 559 calculation of by data acquisition system, (see also Slit flow; Hole pressure; Exit 572 pressure) definition of, 62 Slot casting (see Sheet extrusion) dependence on strain or rate amplitude, Solid body motion, 604 222 Solid body rotation, 604n effect of molecular weight on, 95 Solution viscosity, 613 for linear polymers, 67 Specific viscosity, 613 from modified Rouse model, 77 Spike strain test, 193 low-frequency limiting behavior of, 65 Spiral mandrel die, design of, 538 of cross-linking material, 420 Spiral mold test of typical molten polymers, 66 for cross-linking systems, 414 relationship to molecular weight for injection molding resins, 503, 505 distribution, 68, 96 Squeezing flow (see also Complex modulus) use of to evaluate epoxy resin, 414 Stored elastic energy, 214 (see also Lubricated squeezing flow) Strain, 6, 7, 605 Stability of polymer finite measures of, 108 importance of, 577 nonlinear measures of, 131 measurement of, 578 (see also Strain tensor) Stagnation flow, 545, 546 Strain hardening (see Extension thickening) Start-up of planar extensional flow, 264 Strain rate (see Rate of deformation) Start-up of steady simple extension, 239 Strain softening behavior, definition of, 126 linear response, 74 (see also Extension thinning) of a rubberlike liquid, 124 Strain tensor Start-up of steady simple shear Cauchy strain tensor, 111 linear response in, 72 Finger strain tensor, 111 nonlinear behavior, 194 for simple extension, 35 Steady simple shear for simple shear, 34 definition of, 155 infinitesimal, 31, 33, 34, 105, 111 importance of, 153 of Seth, 130 linear viscoelastic behavior, 47 relative, 112 velocity distribution for, 36 Strength of the network at the gel point, 420 Steady state compliance, 55, 58, 65, 70, 207 Stress, 3, 5 effect of MWD on, 57, 60, 86, 372, 380 (see also Stress tensor) effect of temperature on, 374 Stress growth coefficient (see shear stress effect of branching on, 388 growth coefficient; Tensile stress growth of blends, 372 coefficient) related to molecular weight, 371 Stress growth function (see shear stress related to N!, 121, 374 growth coefficient; tensile stress growth related to plateau modulus, 372 coefficient) related to viscosity, 377 Stress-optical coefficient, 358 from Doi-Edwards theory, 84 Stress overshoot, in steady simple shear, 196 for linear behavior, 56 Stress ratio from modified Rouse model, 77 and the recoverable shear, 122 ofLDPE,254 and ultimate recoil, 214, 216 Steady state tensile compliance, 241 related to parison swell, 519 Step shear in extension, 123 Stress relaxation, 16, 43 Step shear strain, 181, 183 in molten polymers, 50 use of to probe relaxation, 219 of elastomer, 48 (see also Relaxation modulus) (see also Relaxation modulus) Storage and loss compliances, 69 Stress tensor, 25 for typical linear polymer, 70 components, 27 Storage and loss moduli components of simple shear, 27, 154 use of to determine Maxwell parameters, for extensional flows, 238 100 for simple extension, 29 664 SUBJECT INDEX Stress tensor (cont'd) Tensile viscosity in fluid at rest, 30 effect of molecular weight on, 255 meaning of indices, 27 effect of MWD on, 254 relationship to surface stress vector, 26 from entrance pressure drop, 323-324 sign convention for, 27 ofHDPE,258 symmetry of, 28, 154 of LDPE, 251, 255 Stress-optical relation, 357-359 of rubberlike liquid, 124 failure of, 363 use of for quality control, 598 Stretching flow, 232 (see also Apparent extensional viscosity) Strong flow, 20, 150,265 Tensor, 22-23 exponential shear as, 225 as operator, 24 simple extension as, 238 components of, 24 birefringence in, 360 notation for, 25 Structural Time Dependency, 16 Terminal relaxation time, 53 Structure related to steady state compliance, 57 of a melt, 217 Terminal zone, of viscoelastic behavior, 50, of a fluid, 16 58,66 Substantial derivative, 39 Testing Machines Inc., 628 Superposed deformations, 217, 219 Thermal stability, 577 Superposed steady and oscillatory shear, 218 Thermogravimetric analysis, 577 Surface force, 4, 5, 25 Thermorheologically simple behavior, 87 Surface stress vector, 25 Thermosetting materials, 411 Surface tension, effect of on tensile rheological testing of, 414 measurement, 244 Thickness swell (see Parison swell) Suspensions, 17 Thixotropic loops, 221 (see also Filled melts; Viscosity) Thixotropy, 17 Swell (see Extrudate swell) Time constant in viscosity equation, 161-163 Time-temperature superposition, 86, 87, 94, Temperature distribution, in drag flow 96 rheometers, 274 and viscosity function, 169 Tensile creep compliance creep recovery, 213 nonlinear, 240 failure of, 92 of polystyrene, 259 nonlinear viscoelasticity, 188 Tensile flow (see Simple extension) of nonlinear creep time, 209 Tensile recoil, 241 start-up of steady shear, 215 ofLDPE,249 ultimate recoil, 215 prediction of, 257 Time-temperature-transformation, 411 Tensile relaxation modulus, 43, 238 Time-Temperature Instruments, 286, 628 Doi-Edwards prediction of, 148 Tinius Olsen, 628 Tensile start-up flow, 254 Torque rheometers, 287 Te~ile stress,S, 24 use of to study stability, 578 (see also Net tensile stress) Torsional braid analysis (TBA), 416 Tensile stress decay coefficient, 240 Torsional impregnated cloth analysis (TICA), linear behavior, 73 416 Tensile stress growth function Toyo Seiki, 628 for rubberlike liquid, 124, 125 Transducers of LDPE, 251 compliance of, 271 Tensile stress growth coefficient for shear stress, 292, 293 at high strains, 253 Transient shear tests, usefulness of, 228 definition of, 239 Transition zone, of viscoelastic behavior, 50, linear behavior, 73 58 ofHDPE,258 Triple-step shear strain test, 194 of LDPE, 250, 251 Tube flow (see Capillary flow; Channel of polystyrene, 259 flow) related to shear stress growth coefficient, Tube model (see Doi-Edwards theory) 231 Tube renewal, 85 Tensile stress relaxation, 133 Twin-screw extruders, 485-489 SUBJECT INDEX 665 Type I nonlinear behavior, 186 of immiscible blends, 401 Type II nonlinear behavior, 186, 188, 193 of molten polymers, 158 of pre-gel liquid, 418 Ultimate recoil, 57 pressure-independent plot of, 169 during start-up flow, 73 relationship to damping function, 168 following steady shear, 214 shear rate for onset of shear thinning, 377 following steady simple shear, 215 (see also Zero shear viscosity) for a crosslinked material, 123 Viscosity average molecular weight, 611, 615 nonlinear, 211 Viscous dissipation (see Dissipation) of LDPE, 213, 251 Viscous heating of rubberlike liquid, 122 in capillary rheometers, 326 relationship to loss angle, 216 in rheometers, 274 relationship to stress ratio, 216 (see also Dissipation) time-temperature superposition, 213, 215 Viscous stress (Extra stress), 31 (see also Recoil; Tensile recoil; Ultimate Vitrification, during cross-linking, 411 tensile recoil) Voigt body, 13, 48 Ultimate recoil function, 252 Volume fraction, effect on viscosity, 391 nonlinear, 211 Ultimate tensile recoil, 241 Wagner's equation, 130, 193-194 measurement of, 244 and exponential shear, 227 of LDPE, 251 generalization of, 266 Uniaxial extension (see Simple extension) not a constitutive equation, 146 Unit tensor, 111 Wall slip, 158 in capillary flow, 305 related to extrudate distortion, 340 Vector, 23 Weak flow (see Strong flow) Vectra,425 Weber Number, for immiscible blends, 403 Venturi flow, 546, 547 Weight average molecular weight, 608 Viscoelasticity, 42 Weight swell (see Parison swell) of a rubber, 13 Weissenberg effect, 105, 275, 285 Viscometric flow Weissenberg number and low frequency oscillatory shear, 220 for channel flow, 331 examples of, 156 for oscillatory shear, 221 definition of, 155 Weld line Viscometric functions, 155, 195 healing of, 495 Viscosity, 11 in injection molding, 495 definition of, 12, 155 reducing the effects of, 496 dependence of on temperature, 169, 381, role of temperature, 496 582-585 in blow molding, 510 dependence of on molecular weight WLF equation, 89, 382-383, 584 distribution, 374 dependence of on pressure, 384 Yield stress, 18, 397 dependence on shear rate and molecular measurement of, 397 weight, 166 methods of measuring, 286 dependence on shear rate, 159, 374-381 of a suspension, 395, 397, 398 effect of filler concentration on, 390 Yo-yo model, 360 effect of filler particle asymmetry on, 394 effect of filler particle size distribution on, Z-average molecular weight, 609 394 Zero-shear viscosity, 12 effect of filler volume fraction on, 392 dependence of on temperature, 89 effect of molecular weight on, 164-167 from modified Rouse theory, 76 factors it depends on, 159 relationship to relaxation modulus, 73 of a Newtonion fluid, 11 Doi-Edwards theory, 84 of a suspension, 391 effect of molecular weight on, 80, 164-167 of a suspension, effect of shear rate on, of blend, 166 395 related to relaxation modulus, 47 of a concentrated suspension, 392 related to molecular weight distribution, of fiber reinforced melts, 393 368-370