Molecular Weight Distribution of Polymers from Rheological Measurements in Dilute Solution Warner L. Peticolas Citation: The Journal of Chemical Physics 35, 2128 (1961); doi: 10.1063/1.1732219 View online: http://dx.doi.org/10.1063/1.1732219 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/35/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Kinetics of chain collapse in dilute polymer solutions: Molecular weight and solvent dependences J. Chem. Phys. 126, 134901 (2007); 10.1063/1.2715596 On the Rouse spectrum and the determination of the molecular weight distribution from rheological data J. Rheol. 44, 429 (2000); 10.1122/1.551094 More on the prediction of molecular weight distributions of linear polymers from their rheology J. Rheol. 41, 851 (1997); 10.1122/1.550822 Use of rheological measurements to estimate the molecular weight distribution of linear polyethylene J. 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Downloaded to IP: 141.217.58.222 On: Thu, 27 Nov 2014 08:46:26 http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/327320036/x01/AIP-PT/JCP_ArticleDL_101514/PT_SubscriptionAd_1640x440.jpg/47344656396c504a5a37344142416b75?x http://scitation.aip.org/search?value1=Warner+L.+Peticolas&option1=author http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov http://dx.doi.org/10.1063/1.1732219 http://scitation.aip.org/content/aip/journal/jcp/35/6?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov http://scitation.aip.org/content/aip/journal/jcp/126/13/10.1063/1.2715596?ver=pdfcov http://scitation.aip.org/content/sor/journal/jor2/44/2/10.1122/1.551094?ver=pdfcov http://scitation.aip.org/content/sor/journal/jor2/41/4/10.1122/1.550822?ver=pdfcov http://scitation.aip.org/content/sor/journal/jor2/40/5/10.1122/1.550763?ver=pdfcov http://scitation.aip.org/content/sor/journal/tsor/7/1/10.1122/1.548955?ver=pdfcov THE JOURNAL OF CHEMICAL PHYSICS VOLUME 35, NUMBER 6 DECEMBER,1961 Molecular Weight Distribution of Polymers from Rheological Measurements in Dilute Solution WARNER L. PETICOLAS IBM Advanced Systems Development Division, San Jose, California (Received May 29, 1961) A method for obtaining the molecular weight distribution of linear polymers from rheological measure- ments in dilute solution is obtained through the use of the normal-coordinate theory of viscoelasticity as modified by Zimm to include hydrodynamic interactions. Equations for the frequency-dependence of viscoelastic properties in dilute solutions are shown to the form 4>(w) =~pAp-nj(WAp-m), where 4>(w) is a measurable, frequency-dependent viscoelastic property. The summation extends over the set of characteristic values Ap, which accounts for the relaxation of various normal modes of motion of the molecule as perturbed by hydrodynamic interaction with the solvent; and j(WA,-m) is directly related to the molecular weight distribution through an integral equation A,-nj(WA,-m) = l"'K(M, w)W(M)dM, where W(M) is the weight fraction of molecules in range dM at M and K(M, w) is a kernel dependent upon the quantity measured. Inversion of the above summation has been accomplished for the particular case of characteristic values which obey the inequality, ~pAp-n(w). W (M) is then obtained from j(WA,-m) through a solution of the integral equation. INTRODUCTION RECENTLY the author and his fellow workersl- 4 have been devising methods for the experimental testing of the normal-coordinate theory of viscoelas- ticity5-7 and for utilizing the theory to determine the molecular weight distribution of linear polymers. These two aims are not unrelated. It is doubtful that the normal-coordinate theory of viscoelasticity can be adequately tested until the effect of polydispersity is sufficiently well understood. In fact it may be shown that the predictions of the normal-coordinate theory are extremely sensitive to even slight departures from monodispersity. In the past, very few measurements of the visco- elastic properties of dilute solutions of linear poly- mers have been made because of the great experimental difficulties involved in such measurements. How- ever, Birnboim and Ferry8 have recently designed an apparatus which appears to be very successful in measuring the complex viscosity of the dilute polymer solutions. Consequently an extension of our earlier calculations seems desirable and timely. Indeed the results of Birnboim and Ferry show that, in dilute solution, the experimental relaxation distribution func- tions approach a slope of - 2/3 at short times as pre- 'E. Menefee and W. L. Peticolas, Nature 189,745 (1961). 2 J. E. Eldridge and W. L. Peticolas, J. Poly Sci. 51, S38 (1961). 3 E. Menefee and W. L. Peticolas, J. Chem. Phys. 35, 946 (1961). 4 W. L. Peticolas and E. Menefee, J. Chem. Phys. 35, 951 (1961). ⢠P. E. Rouse, Jr., J. Chem. Phys. 21, 1272 (1953). dicted by the theory of Zimm instead of a slope of -1/2 . as predicted by the theory of Rouse and which is generally found in concentrated polymer systems.9 The purpose of this paper is to generalize the previous results1,3 to the case of a nonfree draining polymer molecule in a solution with strong hydrodynamic inter- actions. This problem is more complicated since the characteristic values in this system are neither integers nor simply related to one another. Although the method developed here is applicable to any normal-coordinate treatment with nonintegral characteristic values, only the case of the frequency dependence of the viscosity of dilute solutions of whole polymers will be discussed. COMPLEX VISCOSITY OF WHOLE POLYMERS The pth relaxation time of a polymer molecule containing j monomers in a dilute solution of a whole polymer is given by5,7 (1) where b2 is the mean-square end-to-end length of a submolecule, p is the friction factor, Api is the pth characteristic value of a polymer molecule containing j monomers. For dilute polymer solutions, Zimm has obtained the following equation for the characteristic values: (2) where Ni is the number of submolecules in the Jth polymer molecule, p is the friction factor, Ap is the pth characteristic value as calculated by Zimm, Roe, and Epstein,1O and 7]8 is the viscosity of the solvent. 6 F. Bueche, J. Chem. Phys. 22,603 (1954). 9 J. D. Ferry, R. F. Landel, and M. L. Williams, J. App!. Phys. 7 B. H. Zimm, ]. Chem. Phys. 24, 269 (1956). 26,359 (1955). 8 M. H. Birnboim and J. D. Ferry, Bull. Am. Phys. Soc. 6, 127 10 B. H. Zimm, G. M. Roe, and L. F. Epstein, J. Chem. Phys. (1961). 24,279 (1956). 2128 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.217.58.222 On: Thu, 27 Nov 2014 08:46:26 MOLECULAR WEIGHT DISTRIBUTION OF POLYMERS 2129 The complex viscosity of a polymer in dilute solution is given by the equation5â¢7 00 N 1]*(w) -1].= L)W,/Mi)cRTL:Tp;[1+iwTPiJ-\ (3) ;=1 p=1 where Wi is the weight fraction of the jth polymer molecule, 1]*(w) is the complex viscosity, c is the poly- mer concentration, and i= (-1)!. Combining Eqs. (1)-(3) with the fact I>.p-1= 0.SS6, p=1 we have the following result for the viscosity of a whole polymer at zero frequency, C1]8 W·I\T.J [1] (0) J = N a (1271"3) !0.SS6b3 L:_l' _, j 24M j = 2.S4X 1023 (DIM )w, (4) (S) where L is the root-mean-square end-to-end length of whole polymer chain defined by Li=Ni!b, and iVa is Avagadvo's number. Equation (S) is a statement for whole polymers of Zimm's result for polymer fractions.7 If in Eq. (4) we assume that the number of submolecules LV i is proportional to the molecular weight, i.e., Ni=KMi, we have [1] (0) J = iVa (1271"3) iO.SS6(K1b) 3 (MO.5)w =2.S4X1023 (LNMi)1(Mo.5)w. (6) The dependence of the viscosity of the weight average of the square root of the molecular weight shows that these equations hold only in a 8 solvent,!1 a fact that must be kept in mind when attempting to apply them without further modification. If the viscosity is meas- ured in a 8 solvent, K1b or LNMi is a constant in- dependent of the molecular weightY However, in a non-8 solvent, Klb= (iViIMi)lb is a monotonic, in- creasing function of the molecular weight,!1 so that either iVi/Mi or b or both must be molecular weight dependent. Consequently it would be difficult to extend our discussion of whole polymers to non-8 solvents. Actually, however, there is evidence which would indicate that Zimm's theory itself is more nearly correct for a 8 solvent. Indeed the failure of Zimm's theory to predict a dependence of intrinsic viscosity on steady-state shear rate is not so bad as it might first appear since data on polystyrene in a 8 solvent show only a very slight shear-rate dependence.12 Combining (1) and (6) we have [1](0) J1]8M it (7) 11 P. J. Flory, Principles of Polymer Chemistry (Cornell Uni- versity Press, Ithaca, New York, 1953), p. 612. 12 E. Passaglia, J. T. Yang, and N. Wegemer, J. Poly Sci. 47, 333 (1960). It is interesting to observe that for a given temperature and solvent that the ratio of the intrinsic viscosity to the averaged molecular weight which appears in this equation is a constant independent of both molecular- weight and molecular-weight distribution, i.e., [1](0) JI (MO .⢠)w= [1]i(O) J/Mio.5=K', a constant, which in the Flory terminologyll is K' = if>'(LNM)!, where if>' is the universal constant of Eq. (S) and LNM is independent of molecular weight. A comparison with the corresponding calculation1 for Rouse's theory shows the ratio [1](0) JI (M)w should be a constant in this theory. This is, of course, the result one might expect for a dilute solution of a polymer in an extremely good solvent. Perhaps this is the explana- tion of why the data of Rouse and SitteP3 for poly- styrene in toluene, a very good solvent for polystyrene, agree so well with Rouse's unmodified theory. Equation (7) may be written as where B=1]8[1](0) JI (MO.5 )0.SS6RT. Consequently, Eq. (3) may be written as .V 1]*(w) -1]s= cBR TL:Xp-1I* (WXp-1) , p=l where (S) (9) (10) I*(wX -1) =lOOMo.5W(M)dM (11) p 0 1+iwXp-1BM!' and where we have assumed a continuous distribution of molecular weights. INVERSION OF THE SUM Equation (10) is an equation of the general form N if>(w) = L:xp-nf(Xp-mw) =~NIf(w) l, (12) p=1 where nand m are usually, but not necessarily, integers; the symbol ~ stands for this type of sum. Equations of this form may be inverted to give f(X1-mW ) in terms of an infinite series of if>'s provided the Xp's possess certain properties. If the characteristic values are integers this inversion may be performed using the theorem of Moebius.14 However, the characteristic values of Zimm, Roe, and Epstein10 are not integers and, except at large values, there is no simple relation- ship between them. This may often prove to be the case for problems of this type, so that it seems worthwhile to develop an inversion formula for equations of the form of Eq. (12) which will tend to hold under very general conditions. Such as equation can be developed as shown below. 13 P. E. Rouse and K. Sittel, J .. Appl. Phys. 24, 690 (1953). 14 A. F. Moebius, J. Math. 9, 105 (1832). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.217.58.222 On: Thu, 27 Nov 2014 08:46:26 2130 WARNER L. PETICOLAS For an equation of the form of Eq. (12) we require that MOLECULAR WEIGHT DISTRIBUTION OF POLYMERS 2131 An alternative way of meeting this problem is to use the imaginary part of the complex viscosity (or the real part of the complex modulus) divided by the frequency as follows: 9[ -1/*(w)/wJ=Re[G*(w)/w2J=Hw) Hw) =cB2RTL.Ap-212(WAp-l) , (19) p where j OOM2W (M)dM 12(WAp-l) = 0 l+w2B2Ap-2M3' (20) Before solving for 12 (WAl-l) in terms of Hw) we observe that Eq. (14) may be simplified as shown in the ap- pendix to obtain a sum of N terms rather than an infinite number. Such a sum is still rather tedious to write out so, for the present, we will deal with Eq. (14) symbolically. Consequently, if and if N ct(w) = L.Ap-nf(Ap-mW) = ~N{ few) I ]F'l N L.A i-n < 2Al-n, i=I then the inverse ~-l exists and we may write The exact evaluation of the shortest form for ~-l{ct(W) I will be described in the Appendix. How- ever, if ct(w) is known for all values of w, thenf(Acmw) may be obtained from this equation. SOLUTION OF THE INTEGRAL EQUATION If, in Eqs. (12), and (21) we set m=l, n=2, ct(w) =Hw), Ap-nf(Ap-mW) = Ap- 2cB2RTI2 (WAp-l) , we have the following integral equation: j OOM2W(M) dM CB2RTAC2 =~-l{Hw) I' o 1 +w2B2Al-2M3 ' making the substitution, Z=A12B-2M-3, and y=w2, we have where St is the Stieltjes transform operator and G(z)=(cRT/3z)W(B-2/3A12/3Z-l/3). Using the inverse Stieltjes operator SrI we may write the weight dis- tribution function where M = (Al/wB) 2/3. Using the inverse Stieltjes operator of Hirshmann and Widder15 we obtain 3A -2B-2M-3 dv 1 lim (-I)v(e/v)2v-- cRT noo d(w2)v W(M) X[W4V~~-1{Hw) IJ. (24) d(w2) v For very broad molecular weight distribution v= 1 or v= 2 will generally suffice. For distribution containing sharp curves or corners, this is not a practical method, and numerical methods for the inverse of this integral equation must be found. The type of numerical method selected will depend to some extent upon the type of data under study. An excellent discussion of numerical solutions of integral equations is given by Kopal,l6 One of the simplest numerical methods is the conversion of Eq. (22) into a series of nonhomogeneous simultaneous equations using a quadrature formula to express the integral as a sum. Such a procedure gives the molecular weight distribution from matrix inversion of the simultaneous equations. Sums of the form of Eq. (21) as well as matrix in- versions are easily performed by use of modern high- speed computers. APPENDIX The infinite sum ~-l{ ct(w) I may be written N ct(w) + (-Aln) L.Arnct(Aj-mAlmW) + (-Aln)2 j=2 N N xL. L.ArnAk-nct(ArmAk-mA12mw) k=2 j 2132 WARNER L. PETICOLAS 4 may be grouped together to give N N-l N-2 N-a 4!( -Xln)4L: L: L: L:Xi-nXi-nXk-nXl-n i>3>k>!~2 X(Xi-mXrmXk-mXZ-mXI4fflW) . Similarly there will be only one of each internal term of index 4 with identical indices, N (-Xln)4L:Xi-4n(Xi-4mXI4mw) . i=2 In general there will be an internal term for each partition of the index number, in this case 4. The partitioning of 4 may be illustrated as follows: 1/1/1/1 all different 1 1/1/1 two alike; two different 1 1/1 1 two alike; two alike 1 1 1/1 three alike; one different 1 1 1 1 four alike. The collection of internal terms illustrated above are for the first and last of these partitions. The collection together of all of the internal terms of index 4 in which two indices are alike and two are different gives 4' N N-IN-2 -,-;-!( -X1")4L: L: L:[Xi-2nXrnXk-n 2 .1.1. i>3>k~2 X (Xi-2mXi-mXk-mX14mw) + X i-nX i-2nAk -n eX i-mXr2mXk -mA14",W ) +Xi- nAr"AC2n (Ai-Ai-mXk-2mA14mw )]. (27) It should be noted that the three expressions in the brackets arise from the fact that i>j>k'?:.2 so that A?n can never equal Ar. The internal term for the partitioning into two alike and two alike is given by 4! N N-l 2 '2' ( -Xln)4L: .L:x i-2nXj-2n (X i-2mXj-2mW ). (28) ⢠⢠t>3~2 For the ath term, there will be a collection of internal terms for each of the partitions of a. If these collections of internal terms are all added together, they equal the original sum, but, all of the terms in the new sum are now completely different from each other. This results in an enormous reduction in work necessary to compute ~-11 (w) J. The new terms may be written as follows: 00 N ~-II(w) J = (w) + L: L:( -Xln)vXi-nv v=l i=2 +f f f: f:'E~(_Xlll)v+~+y~V+M+'Y)! v=1 ~=1 'Y=1 k>J>i~2 M!II!'Y! XA.-nvXrnPXcmY (Al",v+mP+mvA i-mvAi-mI'Xk-'lnY) + .. '. (29) The summations over the Greek indices are summations due to the partitions of the index numbers and extend to infinity since the index numbers themselves extend to infinity, while the summations over the italic indices are over the characteristic values and extend only to N. In the double sum we have collected all of the in- ternal terms which result from partitions containing the whole index number. In the quadruple sum, we have collected all of the internal terms which arise by partitioning of the index numbers into two parts with II alike in the first part and p. alike in the second. Simi- larly in the sextuple sum, we have collected all of the internal terms which arise because of the partitioning of the index numbers into three partitions. This process may be continued up to the point where the number of sums over the characteristic values approaches the actual number of the characteristic values. The very last term of Eq. (29) will contain no sums over the characteristic values but only over the partition. This follows from the fact that it is impossible to partition an index number larger than (N -1) into more than (N -1) partitions without some of the partitions containing identical elements. Thus the last term of Eq. (29) consists of exactly N -1 sums and may be written as f f f···:E( -Aln)v+~+Y+ ... +r v=1 ~=1 y=1 ;=1 (V+M+'Y+' â¢â¢ +5") ! X X2-nvxa-nPAr"'Y· â¢â¢ Xx-lli V!M!")'!" 'sl X (Alm(v+I'+Y+ ... +l'lA2-mvxa-mJ.lXrm'Y â¢â¢â¢ XN-mrW). (30) For high-molecular-weight polymers, N is generally taken as infinitely large. For low-molecular-weight polymers N may actually be fairly small. The points made above may be very easily illustrated with N = 3, which might be an actual value for a very short polymer chain. For N=3, (w)=~alf(w)l 3 = L:Ai-"f(Xi-mW) , (31) i~l and co 3 = (w) + L: L:( -Xln)vXi-nv(AlmvA2mvW) p=l i=2 + f f( -Aln) v+p (v+p.) ! ,=1 p~1 1I1p.! xx2-nvxa-nll (Almv+mI'A2-mvA3-mI'W ). (32) Thus the very complicated sum of Eq. (25) containing an infinite number of terms may be replaced with a sum which contains only N terms; in this case N=3. In our previous publication, Menefee and the authorl gave a recursion formula for use in the in- versions of finite sums where the characteristic values were integers. It appears, however, that for finite sums, the approach outlined here might be preferable even when the characteristic values are integers. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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