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A model for the zero shear viscosity.


Zero shear shear: see strength of materials.

A straining action wherein applied forces produce a sliding or skewing type of deformation.
 viscosity, [[Eta].sub.0], being a fundamental property of polymeric polymeric /poly·mer·ic/ (pol?i-mer´ik) exhibiting the characteristics of a polymer.

1. Having the properties of a polymer.

 materials, has been the subject of intensive studies aimed at elucidating its relationship with the polymeric structure (1-5). From experimental and theoretical studies the presence of a double regime of viscosity was quickly ascertained, depending on the molecular weight. At low molecular weights the relationship of viscosity with molecular weight was found to be substantially linear (1, 6), while at molecular weights higher than a critical value, [M.sub.c], the relationship was found to follow the 3.4 power of molecular weight. The 3.4 power law equation was first proposed in 1951 by Fox and Flory (3) on the basis of measurements on narrow distribution fractions of polystyrene polystyrene (pŏl'ēstī`rēn), widely used plastic; it is a polymer of styrene. Polystyrene is a colorless, transparent thermoplastic that softens slightly above 100°C; (212°F;) and becomes a viscous liquid at around 185°C;  and polyisobutylene. It has since been shown to apply to both melts and concentrated solutions for many species of polymers (1, 4).

The presence of a sharp transition between the two regimes at M = [M.sub.c] has been the subject of particular attention since it turned out to be a characteristic constant of the species in the melt state; [M.sub.c] was found equal to about 2-3 [M.sub.e], where [M.sub.e] is the average entanglement molecular weight (1). The sharpness of the transition, however, remained quite puzzling.

The nature of the entanglements has often been discussed and criticized in literature; perhaps, a sounder representation is to consider them as being time fluctuating fluc·tu·ate  
v. fluc·tu·at·ed, fluc·tu·at·ing, fluc·tu·ates

1. To vary irregularly. See Synonyms at swing.

2. To rise and fall in or as if in waves; undulate.

 and rather de-localized (1). More recently a different description of the macromolecular mac·ro·mol·e·cule  
A very large molecule, such as a polymer or protein, consisting of many smaller structural units linked together. Also called supermolecule.
 structure has been proposed, the so-called reptation model, introduced by De Gennes (7) and developed by Doi and Edwards (8, 9), where an entanglement spacing is not specified but an equivalent concept, the tube diameter, has been introduced, which allows an alternative description of the polymeric structure. The entanglements, however, are considered a key factor controlling not only the melt rheology but also the solid mechanical (10-12) and adhesive adhesive, substance capable of sticking to surfaces of other substances and bonding them to one another. The term adhesive cement is sometimes used in place of adhesive, especially when referring to a synthetic adhesive.  properties of polymers (13). For instance, according to according to
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

 Kramer, their relative values determine crazing craze  
v. crazed, craz·ing, craz·es
1. To cause to become mentally deranged or obsessed; make insane.

2. To produce a network of fine cracks in the surface or glaze of.

 versus shear yielding behavior (11, 12).

The increasing evidence of a correlation of material properties to entanglement junctions has stimulated new detailed studies on the correlation between entanglements and molecular structures, resulting, aside basic considerations, in a deep reexamination re·ex·am·ine also re-ex·am·ine  
tr.v. re·ex·am·ined, re·ex·am·in·ing, re·ex·am·ines
1. To examine again or anew; review.

2. Law To question (a witness) again after cross-examination.
 of experimental data and the ensuing en·sue  
intr.v. en·sued, en·su·ing, en·sues
1. To follow as a consequence or result. See Synonyms at follow.

2. To take place subsequently.
 extensive compilations of [M.sub.e] for a broad variety of polymeric species (14, 15).

In this paper we have reexamined the definition of the average number of entanglements per molecule in order to provide a quantitative ground for the zero shear viscosity. It will be shown that a suitable expression for the average number of entanglements per molecule may allow a simple new model for viscosity, which, by the way, quite naturally predicts the features of the sharp transition observed for the relationship viscosity versus molecular weight.


For the moment, we shall restrict our attention to monodisperse A collection of objects are called monodisperse if they have the same size - i.e. their size distribution is effectively a delta function. A sample of objects with a broader size distribution is called polydisperse. In practice, exactly monodisperse collections rarely exist.  polymers.

Let us consider the average number of entanglements per molecule, [n.sub.e]. It can be easily seen from geometric considerations, Fig. 1, that [n.sub.e] increases according to the equation

[n.sub.e] = (M/[M.sub.e] - 1) (1)

which, in the limit of very high molecular weights, tends to M/[M.sub.e]. This function predicts that the number of entanglements per molecule is 1 when M = 2[M.sub.e]; at this level, indeed, it is considered that the entanglement-dominated behavior begins for the viscosity, characterized by the well-known power 3.4.

Equation 1, while adequate to represent [n.sub.e] at high molecular weights, is not fully satisfactory for molecular weights comparable to [M.sub.e]. Indeed, according to this formula, we should expect [n.sub.e] = 0 for M = [M.sub.e] and even negative values for still lower molecular weights, a nonsensical expectation from the physical point of view.

A possible way out for this may be found along these lines:

a) Macromolecules Macromolecules
A large molecule composed of thousands of atoms.

Mentioned in: Gene Therapy

 may obstruct ob·struct
To block or close a body passage so as to hinder or interrupt a flow.

ob·structive adj.
 themselves, when flowing, even at molecular weights equal to or lower than [M.sub.e]. Since the nature of these "effective contacts" or "flow restrictions" is very similar to that defined by the entanglements, we may count as entanglements all the flow restrictions, even those experienced by short molecules;

b) The entanglements do not show up abruptly a·brupt  
1. Unexpectedly sudden: an abrupt change in the weather.

2. Surprisingly curt; brusque: an abrupt answer made in anger.

 at some well-defined molecular weight but appear gradually as a function of molecular weight. Accordingly, we require that the equation defining [n.sub.e] as a function of M must be continuous, starting from zero for M = 0 and converging con·verge  
v. con·verged, con·verg·ing, con·verg·es

a. To tend toward or approach an intersecting point: lines that converge.

 to formula (1) at high M.

A tentative empirical equation satisfying such requirements is the following one,

[n.sub.e] = M/[M.sub.e] [e.sup.-[M.sub.e]/M (2)

This equation may possibly find a theoretical ground; however, without trying to further pursue this goal, we note that it certainly satisfies the above-mentioned requirements, as can be easily seen. Indeed it tends to zero when M tends to zero and to Eq 1 at high M, as can be easily seen by a series expansion of the exponential 1. (mathematics) exponential - A function which raises some given constant (the "base") to the power of its argument. I.e.

f x = b^x

If no base is specified, e, the base of natural logarthims, is assumed.

[n.sub.e] = [M/[M.sub.e] exp exp
1. exponent

2. exponential
(-[M.sub.e]/M) [approaches] M/[M.sub.e] (1 - [M.sub.e]/M) = M/[M.sub.e] - 1. (3)

The use of Eq 2 to estimate the number of entanglements per molecule, here introduced on purely mathematical grounds, entails some significant differences and novelties over the usually accepted estimations of entanglements accounts, such as Eq 1.

First, it describes a continuous development of entanglements over the whole molecular weight range. Second, it satisfies our requirement of avoiding negative values for M [less than] [M.sub.e], which cannot be avoided when using Eq 1. Third, it predicts a number of entanglements differing from zero for M [less than] [M.sub.e]. At this molecular weight, Eq 2 predicts 1/e = 0.37 entanglements per molecule. For M = 2[M.sub.e] Eq 2 gives an estimation of [n.sub.e] of 1.21, a value 21% higher than the classical value of 1 obtained from Eq 2. Finally, as mentioned above, [n.sub.e] estimated by Eq 2 converges to the values estimated by Eq 1 for molecular weights higher than n[M.sub.e], when n is higher than about 4.

In conclusion, we can summarize sum·ma·rize  
intr. & tr.v. sum·ma·rized, sum·ma·riz·ing, sum·ma·riz·es
To make a summary or make a summary of.

 the above considerations stating that Eq 2, which does not introduce new molecular parameters other than [M.sub.e], looks formally adequate to estimate the number of entanglements per molecule over the full range of molecular weights. From the physical point of view, it may be useful to stress again the fact that the above analysis simply means that short macromolecules experience some restriction to their movements even if they do not form a fluctuating network, as implied in the concept of entanglements.


Having a formula for estimating the number of entanglements/molecule, we are now ready to reconsider the dependence of zero shear viscosity, [[Eta].sub.0], on molecular weight for monodisperse polymers. Let's suppose that the viscosity could be written as the sum of two terms: a) the first one, describing the friction between small molecules, depending on the monomeric monomeric /mono·mer·ic/ (mon?o-mer´ik)
1. pertaining to, composed of, or affecting a single segment.

2. in genetics, determined by a gene or genes at a single locus.
 friction factor Friction factor can refer to:
  • Darcy friction factor
  • Fanning friction factor
  • Atkinson friction factor (ventilation of mines)
, [[Zeta].sub.0], and on the total number of chain monomers, M/[m.sub.0]: b) the second one, dealing with the difficulties found by flowing entangled en·tan·gle  
tr.v. en·tan·gled, en·tan·gling, en·tan·gles
1. To twist together or entwine into a confusing mass; snarl.

2. To complicate; confuse.

3. To involve in or as if in a tangle.
 macromolecules, depending on the entanglement friction factor, [[Zeta].sub.e], and on the number of entanglements per molecule, [n.sub.e]; this last term is taken with the 3.4 power to take into account all the available experimental information, which points to such power in the entanglements region. Then we write:

[Mathematical Expression A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  Omitted] (4)

where [[Eta].sub.0] is the zero shear viscosity, [[Eta].sub.0] = M/[m.sub.0] is the number of monomeric units in a chain, i.e., the polymerization polymerization

Any process in which monomers combine chemically to produce a polymer. The monomer molecules—which in the polymer usually number from at least 100 to many thousands—may or may not all be the same.
 degree, and [n.sub.e] the number of entanglements per molecule.

The friction coefficients [[Zeta].sub.0] and [[Zeta].sub.e].sup.3.4], having the dimension of the viscosity, contain, respectively, all the relevant information about the restraints experienced by small molecules moving over the others and the additional restraints to the movement caused by the entanglements. Some comments on them will be made later.

A virtue of Eq 4, Fig. 2, is to be able to describe by a single continuous function the behavior of viscosity of low and polymeric systems over a full range of molecular weights. At low M, i.e., M [less than] [M.sub.e], it converges toward the classical linear relationship, known for low molecular liquids, since the first term is prevailing

log [[Eta].sub.0] = log ([[Zeta].sub.0] M/[m.sub.0]) (M [less than] [M.sub.e]) (5)

and at high M, i.e. M [greater than] 4[M.sub.e], toward the law

log [[Eta].sub.0] = 3.4 log [[Zeta].sub.e] + 3.4 log (M/[M.sub.e] exp (-[M.sub.e]/M)) (6)

which, for M [greater than] 10 Me, further simplifies, reducing to the classical 3.4 power law of high polymeric materials

log [[Eta].sub.0] [Congruent con·gru·ent  
1. Corresponding; congruous.

2. Mathematics
a. Coinciding exactly when superimposed: congruent triangles.

] 3.4 log [[Zeta].sub.e] + 3.4 log M/[M.sub.e] (7)

In the molecular weight region, [M.sub.e] [less than] M [less than] 4[M.sub.e], the two contributions are comparable, so they both contribute significantly to the viscosity.

Accordingly, we may speak of three flow regimes: a) the monomeric regime, up to M [less than] [M.sub.e], where Eq 5 applies; b) the transition regime, for [M.sub.e] [less than] M [less than] 4 [M.sub.e], where the monomeric flow behavior coexists with an incipient incipient (insip´ēent),
adj beginning, initial, commencing.


beginning to exist; coming into existence.
 entanglement-like behavior; and finally c) the high polymer or entanglement regime, M [greater than] 4 [M.sub.e], where Eq 6 applies. Within this region, we can use the approximate law expressed by Eq 7 only when the molecular weight is sufficiently high, i.e., higher than 10 [M.sub.e].

A second characteristic of Eq 4, which clearly shows up from Fig. 3, is that the viscosity shows a rather abrupt slope variation in a narrow molecular weight range, centered around 2[M.sub.e]. This behavior, well known in literature (1), comes out naturally from Eq 4, indicating, as expected, that the entanglement regime becomes really effective on viscosity when, on the average, one entanglement per molecule is present. On this ground, [M.sub.c] can be suitably defined on the basis of the number of entanglements per molecule.

The friction coefficients [[Zeta].sub.0] and [[Zeta].sub.e].sup.3.4], which are characteristic of a given polymeric species and depend on temperature and pressure, can be easily estimated for each polymeric species by Eqs 5 and 7 by testing on low and high molecular weight polymers respectively, as shown in the next section.

As for the estimation of [[Zeta].sub.0], a particular point has to be considered to get meaningful values of it, which was deeply investigated by Graessley and co-workers (16). Since the monomeric friction factor reflects the local chain dynamics, it depends on the concentration of chain ends, the effect of which vanishes with increasing molecular weight. This entails that experimental viscosity values must be corrected in order to establish the real dependence of viscosity from M in the low molecular weight range. Graessley's paper (16) suggests a method to estimate the correction, based on free volume concepts, which results in the following correcting equation

[Mathematical Expression Omitted] (8)

where [[Eta].sub.corr] and [[Eta].sub.exp] are the corrected and experimental viscosities for a low polymer of molecular weight M; [C.sub.1] is the WLF WLF Washington Legal Foundation
WLF Wallis and Futuna (ISO Country code)
WLF Waist Level Finder (camera viewfinder type)
WLF Viva La Figa (MotoGP motorcycle races) 
 constant for the molecular weight M, and [[C.sub.1].sup.varies] is the value of C at high molecular weight. The correction, requiring a preliminary analysis on a set of low molecular weight polymers, is important because only in this way can low polymers be shown to follow the familiar power low pattern, i.e., near to unity, that we have taken for granted Adj. 1. taken for granted - evident without proof or argument; "an axiomatic truth"; "we hold these truths to be self-evident"
axiomatic, self-evident

obvious - easily perceived by the senses or grasped by the mind; "obvious errors"


In order to check the applicability of Eq 4 and to estimate the constants of the model, we have taken into account experimental data from literature on selected well-characterized polymers. Details of characterization, not reported here, may be found in the original papers.

a. The Case of Polybutadiene

We have used the remarkably accurate set of data from Colby, Fetters fet·ter  
1. A chain or shackle for the ankles or feet.

2. Something that serves to restrict; a restraint.

tr.v. fet·tered, fet·ter·ing, fet·ters
1. To put fetters on; shackle.
, and Graessley (17) covering a very extended range of molecular weights, from 1 x [10.sup.3] up to 1.65 x [10.sup.7]. Our analysis, however, was limited to molecular weights up to 350,000, where the 3.4 power was found to apply; above this value, experimental results were found consistent with the lower power of M, about 3, as suggested by the reptation theory. Samples obtained by anionic an·i·on  
A negatively charged ion, especially the ion that migrates to an anode in electrolysis.

[From Greek, neuter present participle of anienai, to go up : ana-, ana-
 polymerization were nearly monodisperse and with similar microstructure mi·cro·struc·ture  
The structure of an organism or object as revealed through microscopic examination.


a structure on a microscopic scale, such as that of a metal or a cell
. [T.sub.g] was found to be -99 [degrees] C and [M.sub.e] = 1850, based on a plateau modulus See modulo.  [[G.sub.N].sup.0] = 1.20 X [10.sup.7] dyn/[cm.sup.2].

In Table 1, zero shear viscosity data are reported at 25 [degrees] C for the molecular weight range where the entanglement regime is supposed to hold, i.e., 10,000 [less than] M [less than] 350,000. In the two last columns are reported the number of entanglements per molecule, [n.sub.e], and the entanglements friction factor, [[Zeta].sub.e], estimated respectively by Eqs 2 and 6. As can be seen, the values of [[Zeta].sub.e] are rather similar, ranging from 0.49 to 0.43. The range could be even a little more narrowed, considering only samples having a well-formed entanglement network, i.e., n [greater than] 20, and excluding sample B3, [TABULAR tab·u·lar
1. Having a plane surface; flat.

2. Organized as a table or list.

3. Calculated by means of a table.


resembling a table.
 DATA FOR TABLE 1 OMITTED] which is in the borderline borderline /bor·der·line/ (-lin) of a phenomenon, straddling the dividing line between two categories.
 of the validity of the 3.4 power law. Accordingly, for polybutadiene at 25 [degrees] C, [[Zeta].sub.e] = 0.445 [+ or -] 0.02 [(Pa.s).sup.1/3.4]. This indicates that a proper choice of [n.sub.e] in the model results in a unique estimation of the entanglements friction factor.

In order to estimate [[Zeta].sub.0] we have taken into account four low polymer samples, Table 2, with molecular weights ranging from 1030 to 1420. Using for the viscosity the already chain ends corrected values, as mentioned at the end of the above paragraph, we got for polybutadiene at 25 [degrees] C: [[Zeta].sub.0] = 0.031 [+ or -] 0.03 Pa.s. Finally, in order to check the validity of Eq 4 we have compared experimental and calculated viscosity data for intermediate molecular weights 1420 [less than] M [less than] 10500, see Table 3 and Fig. 4, which is the transition region from low to high polymer behavior. This region, where the two flow regimes mix together, appears the most suitable for a critical test of Eq 4. As can be seen, estimated viscosity data appear in rather good agreement with the experimental data.
Table 2. Estimation of [[Zeta].sub.0] for Polybutadiene. T:
25 [degrees] C.

Sample      M      corr.[Eta](Pa.s)      [[Zeta].sub.0] (Pa.s)

CDS-B2    1030           0.6                     0.0315
C1        1130           0.7                     0.0335
C2        1190           0.61                    0.0277
C3        1420           0.85                    0.0323

Ref. as in Tab. 1.


To conclude, we have shown in Fig. 5 the comparison of experimental values of viscosity with calculated ones over the whole range of molecular weights.

b. Other Remarks

The example of polybutadiene developed in detail has shown us how to deal with experimental data in order to extract the parameters of the model. For [[Zeta].sub.e] it may be sometimes more straightforward to use the relationship [[Eta].sub.0] versus molecular weight on monodisperse polymers, [[Eta].sub.0] = k [M.sup.3.4], provided that viscosity data were obtained on samples of sufficiently high molecular weight, i.e., M [greater than] 10 [M.sub.e]. Under this condition

[[Zeta].sub.e] = [M.sub.e] [k.sup.(1/3.4)] (9)

where k is the coefficient coefficient /co·ef·fi·cient/ (ko?ah-fish´int)
1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities.

 of the viscosity-molecular weight relationship in the high entanglement range.

Table 4 shows examples of such calculations, based on literature data (16-22). The [[Zeta].sub.e] values were estimated, considering both the M powers reported by the [TABULAR DATA FOR TABLE 4 OMITTED] authors, the [Alpha] values of column 6, and the standard 3.4 power, last column. Usually the resulting values are not too different. In case of too big a discrepancy DISCREPANCY. A difference between one thing and another, between one writing and another; a variance. (q.v.)
     2. Discrepancies are material and immaterial.
, we think it preferable to rely on values obtained from the power estimated from experimenters, since the coefficient k, from which we get the friction factor, and [Alpha] are somehow related. In any case, [[Zeta].sub.e] values for a very broad set of polymers span over two decades with the average value centering around 1 [(Pa.s).sup.1/3.4].

In order to improve our knowledge about [[Zeta].sub.e], it proved fruitful fruit·ful  
a. Producing fruit.

b. Conducive to productivity; causing to bear in abundance: fruitful soil.

 to examine its dependence on temperature. A preliminary analysis, Fig. 6, shows that:

a) [[Zeta].sub.e] has a very similar temperature dependence for a number of polymeric species;

b) The temperature dependence is a WLF-type, i.e., it can be rationalized by T-[T.sub.g].

These statements may help to predict viscosity, since [[Zeta].sub.e] can be easily estimated at any given temperature from plot 6.

As for [[Zeta].sub.0], the above analysis on polybutadiene has indicated that, before getting acceptable results, it is necessary to make relevant corrections to experimental data. This would require a more specific study and will be postponed to a later time. For the moment we simply note that, at the same temperature, [[Zeta].sub.0] values for polybutadiene turn out to be about half the value of [[Zeta].sub.3.4].

It may be interesting to observe that the present model considers the friction factors as constants, independent from the molecular weight. This is in contrast to what is stated in the literature (Ref. 1, Ch. 10, Eq 14), where the monomeric friction coefficient is assumed to rise from the value [[Zeta].sub.00], for the monomer monomer (mŏn`əmər): see polymer.

Molecule of any of a class of mostly organic compounds that can react with other molecules of the same or other compounds to form very large molecules (polymers).
 to an equilibrium value [[Zeta].sub.0] at high molecular weights, as a consequence of the additional free volume associated with molecular ends. Our approach, Eq 2, explains the small deviations from linearity of the viscosity of low polymers as due to additional entanglement-type constraints to flow found from small molecules even below 2 [M.sub.e].


We can summarize the above considerations along these lines:

The average number of entanglements per molecule, [n.sub.e], was critically reexamined, resulting in a new equation, Eq 2, which applies to monodisperse polymers. The modifications introduced for [n.sub.e], although apparently minor from the quantitative point of view, offer in principle a means to describe by a unique continuous function the evolution with the molecular weight of the number of entanglements per molecule.

The analytical equation for [n.sub.e] was then used to get a description of the zero shear viscosity over an extended range of molecular weights, from very low polymer up to the molecular weights where the 3.4 power law applies. Two well-known experimental findings turned out to be properly described: the continuous evolution of the viscosity as a function of molecular weight, and the sharp slope variation when changing from the monomer friction regime to the high polymer, i.e., to the entanglement regime.

The parameters appearing in the new viscosity equation, Eq 4, are [m.sub.0]. the monomeric molecular weight, [M.sub.e], the average molecular weight between entanglements, [[Zeta].sub.0], the monomeric friction coefficient, and [[Zeta].sub.e], the entanglement friction factor. In particular we underline underline

an animal's ventral profile; the shape of the belly when viewed from the side, e.g. pendulous, pot-belly, tucked up, gaunt.
 here the novelty of the introduction of the entanglement friction coefficient, which accounts for additional restraints experienced by a macromolecule macromolecule, term that may refer either to a crystal such as a diamond, in which the atoms are identical and held by covalent bonds (see chemical bond) of equal strength, or to one of the units that compose a polymer.  when moving within the network of randomly distributed macromolecules. The friction factors depend on temperature and pressure. Eqs 5 and 7 may be used for a quantitative evaluation.

Whereas [[Zeta].sub.0] was estimated only in the particular case of polybutadiene, leaving a deeper analysis to further studies, Se values for many polymers were shown to follow a WLF-type equation, i.e., with quite similar values when referred to temperatures equally distant from [T.sub.g].

Finally, using the friction factors extracted from experimental data at low and high molecular weights, the validity of the general viscosity equation, Eq 4, was tested in the most critical molecular weight region, i.e., for [M.sub.e] [less than] M [less than] 4[M.sub.e]. finding good agreement with the experimental data.


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natural philosophy, physics - the science of matter and energy and their interactions; "his favorite subject was physics"
 Properties of Polymers, 3rd Ed., John Wiley John Wiley may refer to:
  • John Wiley & Sons, publishing company
  • John C. Wiley, American ambassador
  • John D. Wiley, Chancellor of the University of Wisconsin-Madison
  • John M. Wiley (1846–1912), U.S.
 & Sons, New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
, (1980).

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a North American term commonly used to describe heifers close to term with their first calf.
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1. physical

2. physician

3. physiological

4. physiology
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1. the property of remaining in close proximity.

2. the stable joining of parts to one another, which may occur abnormally.

, Marcel Dekker Marcel Dekker is a well-known encyclopedia publishing company with editorial boards found in New York, New York. They are part of the Taylor and Francis publishing group.

Initially a textbook publisher, they went to encyclopedia publishing in the late 1990's.
, New York (1982).

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Author:Locati, G.; Pegoraro, M.; Nichetti, D.
Publication:Polymer Engineering and Science
Date:Apr 1, 1999
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