Structure and Rheology of Molten Polymers

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1 John M. Dealy Ronald G. Larson Structure and Rheoloy o Molten Polymers From Structure to Flow Behavior and Back Aain Sample Chapter : Structure o Polymers ISBNs HANSER Hanser Publishers, Munich Hanser Publications, Cincinnati

2 Structure o Polymers This chapter introduces concepts and models that are used in subsequent chapters o this book. A much more thorouh treatment o polymer structure can be ound in the monoraph o Graessley [1]..1 Molecular Size.1.1 The Freely-Jointed Chain I we know the molecular weiht o a linear polymer, it is easy to calculate the stretched-out lenth o a molecule. However, this dimension is very much larer than the size o a coiled-up molecule in a solution or melt. And it is essential to our purposes to establish a quantitative measure o the size o such a coil. Due to Brownian motion, a polymer molecule is constantly explorin a very lare number o possible conormations due to its reat lenth and lexibility. A detailed analysis o all these conormations would be an enormous undertakin, but i we are interested only in certain averae quantities, such an analysis is not necessary. In act, it is possible to derive some useul expressions by analyzin a reely-jointed chain rather than the actual molecule. O course the sements o the molecule that consist o atoms and chemical bonds do not constitute a reely-jointed chain because o limitations on bond anles and orientations, but on a scale that is somewhat larer than that o a chemical bond but still much smaller than that o the stretched-out molecule, there is suicient lexibility that the molecule does, indeed, act like a reely-jointed chain. In addition to the assumption o a reely-jointed chain, in the ollowin discussion, we will inore restrictions on molecular conormation due to the inability o two sements o a molecule to occupy the same space. A chain or which this is allowed is called a phantom or host chain. Finally, we will assume that the chain is not stretched very much. To summarize, in the ollowin development or the reely-jointed chain we will make use o three assumptions: 1) the molecule is very lon; ) it is a phantom chain; and 3) the chain is not extended by low or external orces. Assumptions two and three imply that the molecule is in an unperturbed state, i.e., that it is ree o the eects o external orces resultin rom low or solvation. These assumptions are applicable to a molten polymer and to a very dilute solution when the combination o solvent and temperature is such that the conormation o the polymer molecules is unaected by polymer-polymer or polymer-solvent interactions, i.e., such that the solution is in its theta state. The theory o the reely-jointed chain is described only in eneral terms in the ollowin section, and a more detailed discussion can be ound, or example, in the text o Boyd and Phillips [].

3 8 Structure o Polymers [Reerences on Pae 5].1. The Gaussian Size Distribution In order to calculate the coil size o a reely-jointed, phantom chain, we start with the assumptions noted above and consider a chain consistin o N reely-jointed sements o lenth b. Since there are no restrictions on the orientation o one sement with respect to its neihbors, the position o one end o the chain relative to the other is iven by a threedimensional random-walk calculation, also called a random-liht calculation. Such calculations can be used to determine the averae end-to-end distance, i.e., the root-mean-square end-to-end vector o a molecule, R, where the subscript indicates that this averae applies to the unperturbed molecule (equivalent to a dilute solution in the theta state deined in Section.1.3) or to a melt. For a vinyl polymer with a deree o polymerization o 1, the root-mean-square end-to-end distance is about nm. Fiure.1 shows a reely-jointed chain superposed on the molecule that it represents. By Brownian motion, a molecule will move throuh a broad samplin o all its possible coniurations in any siniicant period o time. Assumin that each coniuration is equally probable, random liht calculations show that when N is lare, the mean-square end-to-end distance is iven by Eq..1. R = b N (.1) Fiure.1 Sement o a polyethylene molecule with vectors showin a reely-jointed chain that can be used to simulate its behavior. In the chain shown there is about 1 sement or every ten bonds, and each sement is about eiht times as lon as a PE bond. From Boyd and Phillips [].

4 .1 Molecular Size 9 Values o this parameter or several polymers are tabulated in Appendix A. The radius o yration R o a molecule is the root-mean-square distance o mass elements o the chain rom its center o ravity. (It is also the radius o a body havin the same anular momentum and mass as the molecule but whose mass is concentrated at the radius, R.) Averaed over all possible conormations o the reely-jointed chain, the mean-square radius o yration is iven by Eq R = b N = R (.) 6 6 It is also possible to calculate the distribution o end-to end vectors or a random walk, and while the result is rather complex, it is very closely approximated by Eq..3: 3 PR ( ) = 3 π N b 3/ 3 R /N b e (.3) where P(R)dR is the raction o all possible random lihts havin end-to-end radii between R and R+dR. The unction deined in Eq..3 is called a Gaussian distribution, and a molecule in which the end-to-end distance ollows this distribution is oten called a Gaussian chain or a random coil. We note that this probability density only approaches zero at very lare values o R, while in reality the maximum extension o the chain is limited to a inite value. This reminds us that the Gaussian (reely-jointed) chain model is not valid or a hihly extended molecule. Note that the mean-square end-to-end distance has no meanin or a branched molecule, while the mean-square radius o yration is still a meaninul measure o size. We can now use the Gaussian distribution to recalculate the mean-square end-to-end distance: R = R P( R)dR = N b (.4) Thus, the approximate distribution unction o Eq..3 leads to the correct mean-square value. The useul results that arise directly rom the reely-jointed chain model o a polymer molecule are the relationship between the averae size parameters, R and R, and the Gaussian distribution. Now we want to relate the averae size parameters to those describin the chemical bonds makin up the backbone o the actual molecule and thus to its molecular weiht. At the level o the carbon-carbon bonds, the chain is not reely-jointed, as the relative motions o the bonds are limited by the bond anle and rotational enery potentials. One maniestation o this is that the ully-extended lenth or contour lenth o the molecule is less than n l, where n is the number o backbone bonds, and l is the bond lenth. L = Extended (Contour) Lenth = K eom n l (.5) where K eom is the sine o one hal the bond anle, which or polyethylene is 19.47, so that K eom is equal to.816. Another maniestation o the limitations on motions in the actual molecule is that the mean-square end-to-end distance R, which is equal to N b or the

5 1 Structure o Polymers [Reerences on Pae 5] reely-jointed model chain, is considerably reater than n l. The ratio o R to n l is thus a measure o the lexibility o the chain. This quantity, the characteristic ratio, C, can be calculated rom the chain valence anles and the distribution o bond rotational enery states, and it is a constant or a iven polymer. C R nl (.6) The ininity subscript indicates that this value applies when n is suiciently lare that the ratio is independent o n. For polyethylene C is equal to 6.8, while or polystyrene, a stier molecule, it is 9.85 [3]. Values or several other polymers are tabulated in Appendix A. We will let N be the deree o polymerization, which is M/M, where M is the molecular weiht o the polymer, and M is the monomer molecular weiht. The number o bonds in the backbone, n, is then j N, where j is the number o bonds per monomer unit. For vinyl polymers j =. Usin these symbols, the mean-square end-to-end distance can be written in terms o the molecular weiht as shown by Eq..7. R = C nl = C j N l = ( C l j/ M ) M (.7) This shows that R is proportional to the molecular weiht and that R / M is thus a constant or a iven polymer. Combined with Eq.. it shows that the unperturbed meansquare radius o yration is also proportional to the molecular weiht: R = ( C l j/6 M ) M (.8) Several additional characteristic lenths will also be used in this book. One is the eective random-walk step, b n, which is deined by use o Eq..1, with the number o reely-jointed sements set equal to the actual number o backbone bonds, n: n R = nb (.9) so that b n is deined by Eq..1: bn R / n (.1) Thus, b n is the bond lenth o a hypothetical, reely-jointed molecule o n sements that has the same value o R as the actual molecule. Another lenth closely related to b n is the structural lenth introduced by Ferry [4, p. 185]. This is the statistical sement lenth or which we will use the symbol b (Ferry uses the symbol a). This is deined in a similar manner as b n, but with n replaced by N, the deree o polymerization: / / n b R N = R j n = l C j = b j (.11)

6 .1 Molecular Size 11 where j is the number o backbone atoms per monomer unit. Note that when the monomer is an alkene with a sinle double bond or a vinyl monomer, j =, and b = b n. The persistence lenth, L p, is the distance alon the molecule at which the orientation o one sement loses its correlation with the orientation o another. In other words, a bond located a distance L p rom a second bond experiences neliible eect on its orientation due to the second bond. Quantitatively it is deined as the averae projection o the end-to-end vector o an ininitely lon chain in the direction o the irst sement. Doi and Edwards [5, p. 317] show that or a Gaussian chain, this lenth is related to R and the contour lenth L as ollows: L p = R L (.1) Yet another lenth parameter that will be useul is the Kuhn lenth, b K. Kuhn [6] imained an equivalent reely-jointed chain that has the same extended lenth L as the actual molecule. Thus, i the equivalent chain has N K sements o lenth b K, and, K K = = n b R C nl (.13) Rmax = L = NK bk = Keom nl (.14) Where K eom is a constant or a iven chain structure, as explained below Eq..5. In the remainder o the book, R max will be reerred to simply as L. Thus the Kuhn lenth is iven by: bk C nk n = = l N N k eom k (.15) For polyethylene, usin the values mentioned above, b K /l 8, and N K /n 1/1. We also note that the persistence lenth L p is just one hal the Kuhn lenth. The above discussion o molecular size applies to linear molecules. The picture is considerably more complicated or branched molecules. One measure o the eect o branchin on the size o a molecule is the branchin actor, which relects the eect o branchin on the mean square radius o yration or a iven molecular weiht: R R B L (.16) The radii o yration in this equation reer to a molecule in solution in its unperturbed state, and as we will see shortly, they also apply to molecules in the melt.

7 1 Structure o Polymers [Reerences on Pae 5] To relate to parameters describin the branchin level, inormation about the type o branchin present is required. Such inormation can be developed rom knowlede o the polymerization process. (Note that R has no meanin or a branched polymer.) Small [7] lists sources o ormulas ivin or a number o well-deined branchin structures, and we present a ew examples here. For star molecules with arms o equal lenth, Zimm and Stockmayer [8] showed that is iven by: 3 = (.17) For randomly-branched molecules o uniorm molecular weiht, each with n branch points havin a unctionality o, Zimm and Stockmayer [8] made several simpliyin assumptions to arrive at the expressions or (n) shown below as Eqs..18 to. or one, two and three branch points per molecule (n) respectively. 6 (1) = ( + 1)( + ) 3(5 6 + ) () = (4 1) 3(13 + 8) (3) = (9 9 + ) (.18) (.19) (.) For heteroeneous polymers with larer, uniorm numbers o randomly distributed branch points per molecule with a random distribution o branch lenths, they derived Eqs..1 and. or tri- and tetra-unctional branch points, respectively: 1/ 1/ 1/ n + + w ( nw) nw 3 w = ln 1 1/ 1/ nw nw ( + n w) nw (.1) 1 4 w = ln(1 + n w ) n w (.) In these equations, n w is the weiht averae number o branch points per molecule. Lecacheux et al. [9] reported that or n reater than 5, the ollowin approximation is accurate to within 3%. 1/ 3 π 5 = n n (.3)