NOTES
August, 1.962
CALCIJLBTIONS OF ENTA4NGLEMENT COUPLIh’C SPACINGS Ih’ LINEAR POLYiMERS UY
HERSHFL ~ I A R K O VT H IT OM Z4,S~ FERRY
FOX,1 A S D
JOHK D.
UeZZon Instztrtte, Pzttsburyh, Pennsylvanza und Department of Chemzst r y Unzverszty of Wzsconszn, ;Madason, U’zseonszn ReCEzved March 84, 1968
Several different quantities and symbols have heen introduced in the literature to characterize the densit,y of points of entanglement (or coupling) in a linear amorphous polymer of sufficiently high molecular weight. Their definitions have been based 011 different viewpoints which are in some respects iiiconsistent . With the hope of reducing the attendant confusion, we summarize here the various practices which have arisen and propose a uniform usage. In all cases, the calculation of the threshold for an entanglement network has been based on the theory of gelation due to a cross-linking process,2 as though thch coupling loci were equivalent to crosslinks. Gelation theory shows that an infinite network mill be formed from a monodisperse system if one cross-lznkage is introduced on the average for two primary molecules; equivalently stated, since there is one cross-linked u n i t on each of the joined molecules in the cross-linkage, there is an average of one cross-linked unit on each primary molecule. I n a polydisperse system, the criterion is one cross-linked unit per primary molecule on a ~ ~ G g h t - a v ~ r basis. age The cross-linking is, of course, assumed t o be random. Methods for Characterizing Entanglement Spacing. A. From Viscosity-Molecular Weight Data. ‘I’hri viwosity for a homogeneous series of flexible chaiii molecules in bulk or a t a fixed diluent cwicentration is observed to increase with the 3.4 power of the molecular weight *J!l providing 144 2 12,. Abruptly for d/i 5 M,, a less severe dependence of the viscosity on the chain length is found. The observed critical molecular weight, designated as &ifc,is a characteristic of the polymtr’s chemical structure. The corresponding chain length at its “break point,” designated as Z,,is equal to naMc/X0, where J40 is the molecular weight of the monomer unit and n, is the number of chain atoms per uait. The entanglement spacing ha5 been variously computed from AI, as follows. 1. The molecular weight si, lhe break poilit (&Ic) was idtmtified dzrectly with the average molecular weight between entanglements. The symbol IC was used for its re~iprocal.~Accordingly, the observed Zc at the break point was identified with the average number of chain atoms between entaiigleinents.
2 . On the basis of the hypothesis t h d there exist both retarding and accelerating elements,5 the observed value of Mc a t the break point was (1) The suppar” of the Office of Naval Reaearch 1s acknowledged. (2) P J Flow, ‘Principles of Polymer Chemistrj ” Cornell TJnivosit% Presq IthACa N 1 1 ( I ) 1‘ Iluc~lic.7 Chem I’hys , 20, I959 (1952) (4) T G Fox a n d S. Loaliack, J A p p l P h y s , 26, l O b O (1Y;zj (3 I? Bueche a b z d , 26, 738 (1955).
1567
identified as 2Me, where Me denotes6 the average molecular weight between entanglements. Such a factor of 2 has alternaOively been introduced on the hypothesis that the entanglements which change the character of t,he viscosity-molecular weight relationship a t M , must be located near the middle of the chain.7 I n this assignment, t~heaverage degree of polymerization between eiiOanglement points, denoted6 as Z, is M,/Mo. Thus, for polymers with two chain atoms per monomer unit 2, in (Al) is four t’imes2, defined here. B. From the Pseudo-equilibrium Modulus or Compliance.-Another method for estimating i,hc en t!aiiglement spacing involves the value of Lhe storage modulus G’(w), the relaxation modulus Gjt), the storage compliance J’(w), or the creep compliance J ( t ) ic the region of the time or frequency scale where the slope is very small and undergoes an inflection. Taking this value as a pseudo-equilibrium modulus, Gel or compliance, Jq one can calculate an average molecular weight, iWe’, or degree of polymerization, Z,’, between coupled units from rubber elasticity theory: Me’ = Ze‘iW, = pRTJ, = pRT/Ge, where p is the density, R is the gas coiistant, and T i s the absolute t e r n p e r a t ~ r e . There ~ ~ ~ is no uncertainty involving a factor of 2 in these calculations. However, there is usually considera.ble doubt in selecting a suitable magnitude of G, or Je. C. From the Loss Compliance.-The posit,ion and T-alue of the maximum of the loss compliance, J ” ( w ) , also have been employed6J to estimate the enBaiiglement spacing. The values obtained here involve assumptioils similar to those used in treating the dependence of viscosity on molecular weight. Comments and Recommendations.-Thert: arc two import,aiit, inconsistencies in the refereiices cited. One is arbit’rary: the use of the symbol Z to denote nuniher of chain atoms4 aud altermtively degree of polymerization.6 There is a difference of a fact’or of 2 in vinyl polymers, 3 in polyethylene oxide, 4 in Hevea rubber, and larger factors in polyesters. Since the number of chain atoms is the more useful basis of comparison among different polymers, we propose to ezpress spacings o n this basis in the future, but adopt a new symbol il for the average number of chain bonds between coupling entang1emen)ts. The second inconsistency, a more basic discrepancy, involves the question of whether the break point in the dependence of viscosity on molecular weight should be identified with d or 24. From the analogy with gelation theory, the reasons preT4ously given for including a factor of 2 do not appear to be valid. Accordingly, pending the development of rigorous theories and/or a comprehensive body of ,experimental data relating h to the observed M , in the viscosity-molecular weight relation, to the aforementioned inflection point in (6) J. D. Perry, “Viscoelastic Properties of Polymers,” .Jolrn Wiley and Sons, New York, N. T., 1961, p. 287. (7) R . S. Marvin in J . T.Bergen, ed., “Visooelasticity: l’henoinenological Aspects,” Academic Press, New York, N. Y . , 1950, p. 18. ( 8 ) H, Mark and A. V. Tobolsky. “Physical Chemistry of High Polymer Systems,” Interscience Publishers, Kew Yo,-k, N. Y . , 1950, 17. 311. (9) J . D. lt’eny, It. F. I ~ i i d c l nnd , hl. L. N*iltiaiiis, J . A g p l . l’hus., 26, 339 (1955).
NOTES
1568
Vol. 66
TABLE I REVISED srALUES OIi' h,THE AVERAUE E N T A N G L E M E N T Dependence of q
on M u
Polystyrene
730(f, 217")"
Polyvinyl acetate Polyisobutylene
680(f, 160°)d 608(f, 217')"
Hevea Polyethylene oxide Polydimethyl siloxane Polymethyl methacrylate
95O(f, 25')"
SPACING, I N CHAIX A r O M S
Inflection of vkcoelastic propertiesW
Max. in J"m
6 i o ( ~W, , 1000, 3.5 x 103)~ 3 2 0 ( ~f,, i100,3.5 x 1 0 3 ) ~ 338(~, f, 500, 1.8 x 1 0 4 ) ~ 240(E, w, -30", 2.4 X 105)Q 320(E, W, -30", 5.6 X 320(J', w, 90°, 5.6 X IO4)' 296(G', r, -30°, 6 X IO4)' 120; 400((7', W, -30" - ) h 480(0, W, -, -)' 200(J', w, 70°, 7.8 X lo4)"' 320(J', w, 25", 1.3 X 105)" 88(0,f, 140", 8.2 X 104)0 74(E, -, 135", -)"
480
500 272
Jz
1040 (90")f
820 (50")4 400 (-50")'
Polymethyl acrylate 400 760 (f, soo,5.3 x 10419 Poly-n-butyl methacrylatc 368 640 (f, 1250,4.3 x 104)r Poly-n-hexyl methacrylate 720 io80 (f, iojo,4.7 x 104)' Poly-n-octyl methacrylate 1320 2320 ( f , go", 3.8 X lo4)' Poly-n-docosyl methacrylate 5000 19200 ( ~ , 6 0 O , -)" a T. G Fox and S. Loshaek, J . Appl. Phys., 26, 1080 (1955). b H. Fujita and K. Ninomiya, J . Polymer Sci., 24, 433 (1957). A. V. Tobolsky and K. Murakaini, ibid., 40, 443 (1959). T. G Fox and H.. Nakayasu, unpublished. e K. Ninoniiya, J . Colloid Sci., 14, 49 (1959). f M. L. Williams and J. D. Ferry, ibid., 9, 474 (1954). R. D. Andrew and E. R. A. V. Tobolskv, J . Polymer Sei., 7, 221 (1951). E. Cat,siff an'd A. V. Tobolsky, J . Colloid Sci., 10, 375 (1955). Fitzgerald, L . b . Grandine, Jr., and J. D. Ferry, J . A p p l . Phys., 24, G50 (1953). j L;, J. Zapas, S. L. Shufler, and T. W. DeTVitt, J . Polymer Sei., 18, 250 (1955). IC A. R. Payne, in "Rheology of Elastomers, ed. by P. Mason and N. Wookey, F. Pergamon Press, London, 1958, p. 86. There appear to be two distinct pseudo-equilibrium moduli in the data. T. P. Yin, S. E. Lovell, and J. D. Ferry, J . P h y s . Chem., 65,534 (1961). D. J. Bueche, J . Polymer Sci., 25,305 (1967). Plazek, W. Dannhauser, and .J. D. Ferry, J . Colloid Sci., 16, 101 (1961). F. Bueche, J . A p p l . P h y s . , 26, 738 (1955). p J. R. McLoughlin and A. V. Tobolsky, J . Colloid Sei., 7, 555 (1952). q M. L. Williams and J. D. Ferry, ibid., 10, 474 (1955). W. C. Child, Jr., and J. D. Ferry, ibid., 12, 327 (1957). * W. C. Child, Jr., and J. D. Ferry, ibid., 12, 389 (1957). ' W. Dannhausor, W. C. Child, Jr., and J. D. Ferry, ibid., 13, 103 (1958). P. R. Saunders and J. D. Ferry, ibid., 14, I n parentheses we have indicated type of sample and temperature of measurement. uI In parentheses we 239 (1959). have indicated which property, type of polymer, approximate temperature where inflection was measured, and weightor viscosity-average number of chain atoms in polymer sample used. Where no entry appears in another column, we indicate type of polymer sample, temperature, and average chain length. Type of polymer: w, whole; r, rough fraction; f, fraction. Viscoelastic property measurement: E, longitudinal relaxation modulus; J ' , storage shear compliance; G', storage shear modulus; D, longitudinal creep compliance.
'
%
Ihe modulus or compliance, and to the maximum in the loss compliance, we suggest that all lengths calculated by (A2) and (C) above should be multiplied by 2; it follows that no correction of this sort is required in the length computed by (Al) and (B). I n accord with these considerations, we identify A with 2, in (Al), with 2n,Ze in (A2), with naZe' = n,pRT/MoGe in (B), and with 2n,Z, in (C). We further recommend that the method of computation of A be given explicitly in all cases, together with other pertinent details such as the temperature of measurement and information characterizing the molecular weight, naolecular weight distribution, degree of branching, and stereochemical struclure of the polymer samples o n which the data were obtained. Some values recalculated on this basis from a recent compilation, '0 together with other recent literature values, are given in Table I. The data on the dependence of the zero-shear viscosity on molecular weights all mere obtained on fractions a t the temperatures indicated. The viscoelastic function whose inflection point was observed is indicated in the next column. Here most of the data were obtained on unfractionated polymers, (IO) 3. D. Ferry, ref. 0, p. 289.
and for each the viscosily- or wight-average number of chain atoms is indicated if known. 111though the point of inflection frequently is determined from a reduced plot, we have indicated here a temperature in whose neighborhood the actual data were obtained. I n most cases the curves do not have a point of zero slope so that the choice of a value of J , is a t best subject to considerable unccrtaiiity and sometimes may be unjustified. The method employing the maximum in J"(o) involves the same data as used in the inflection point method a t a somewhat different temperature where both kinds of information are available; otherwise more detailed information is indicated. The values of A in Table I are generally of the order of lo2 to lo3. Values obtained by different methods on the same type of polymer generally differ by no more than a factor of three; in some cases the agreement is much better. It is clear that a better theoretical basis for these methods as well as accurate evaluations of A a t the same temperature by all three methods on the same series of polymers covering a range of J!l and/or of molecular weight distribution are required before final conclusions can be drawn concerning their interrelationships.
