Radius of Gyration of Stiff Chain Molecules as a Functionof the Chain Length and the Interactions with the SolventBY R. G. KIRSTEInstitute of Physical Chemistry, University of MainzReceived 8th December, 1969The radius of gyration and the second osmotic virial coefficient of a model chain molecule iscalculated numerically with a Monte Carlo procedure. The model is intended to resemble cellulosetrinitrate (CTN). The expansion coefficients of stiff chain molecules such as CTN or DNA are nearunity at high molecular weights and in good solvents.In fig. 1 the root-mean-square radius of gyration (r2)* of cellulose trinitrate (CTN)in acetone is shown as a function of the molecular weight M. The nitrogen content ofthe CTN was 13.9 %. The results are obtained by light scattering 1 s 2.As can beseen ( r 2 ) is nearly proportional to M. This occurs though the exponent in theviscosity relation [q] = KM" is exceptionally high and the solution is exothermic. Inorder to understand this result, ( r 2 ) and the second osmotic virial coefficient A2 of astiff chain model were calculated numerically as a function of the chain length and thedepth E of a square-well potential. E was varied widely so that the whole range ofpolymer-solvent interaction was covered.MFIG. 1.-Radii of gyration of cellulose trinitrate containing 13.9 % nitrogen in acetone at t = 25"C,r and M are weight-averaged, r is the root-mean-square ; 0, light-scattering measurements. Theslope of the line is 0.51 at the left and 0.53 at the right end.DESCRIPTION OF THE MODEL ( F I G .2)Nvectorsof length I are tiedtogether. Eachofthesevectorsformswithitsneighboursthe angle a, the cosine of which was put constant and equal to 19/21. The positionof a vector relative to the next but one depends in addition to a on the angle of rotation(fig. 3). The N-2 angles of rotation were determined by random numbers. It552 RADIUS OF GYRATION OF STIFF CHAIN MOLECULESmust be emphasized that a and q are not real valence angles and angles of rotation.They are only building materials of the model chain. With the use of a free-rotatingvalence-angle chain as a simple method of chain construction a stiff chain can beobtained only with a valence angle not far from zero. The statistical chain element ofour chain isl + cos a 1+19/211 - cos a ‘I- 19/21 = 201 -b = l The ends of the threads and the joining points of two successive vectors were chosenas centres of a square-well potential with spherical symmetry:00 if x < x10 if x > x2.E if x1 < x < x2It was assumed x1 = 0.6 I and x2 = 0.9 1.The potential u is related to the approach-ing of a second segment. In this respect it is irrelevant whether the two segmentsFIG. 2.-Projection of a random conformation of the used chain model, I = 17 8, is the distancebetween two potential centres, x1 = 0.6 I and xz = 0.9 I indicate the potential for the approachingof a chain fragment A to the chain fragment B (eqn (2)).(A and B in fig. 2) belong to the same chain or not. For the correspondence to theCTN-chain 1 = 17 A was chosen.This value was found in the following way.From the measurements which are shown in fig. 1 the unperturbed dimensions wereestimated by an extrapolation pr0cedure.l’ The statistical chain element of theseimagined unperturbed CTN-chains was found to be 340 long. Therefore we haveto take I = 340/20 = 17 A. The molecular weight of a 340 A fragment of the CTN-chain is = 19 600. The molecular weight belonging to a potential centre of themodel chain therefore emerges as MI = 980. Now the model is entirely defined.PERFORMANCE OF THE CALCULATIONLet us start with the equation 6-8A2 = - N/2 YMzJF1(1)F1(2) [expi - U(l,Z)/kT] - 11 d(1) d(2) (3R. G . KIRSTE 53Nis the Avogadros number and Vis the volume of the system.Fl(i) is the probabilityfor the conformation i, where i is representing all the coordinates which are necessaryto describe the position and the conformation of the molecule. U(1,2) is the intermol-ecular potential energy of a pair of molecules with the position- and conformation-coordinates 1 and 2. d(i) is the product of the differentials of the coordination i.FIG. 3.-The principle of thread construction, o( = valence angle, 'p = angle of rotation.In eqn (3) the integration over the coordinates s1 of the centre of mass of the firstmolecule yields V. Withone obtains thenFl(i) = exp { - U(i)/kT)/Jexp { - U(i)/RT)d(i- Si) (4)U(i) is the intramolecular potential energy and (i - s3 are the coordinates of conforma-tion and position without si.In order to calculate U(l), U(2) and U(1,Z) a model must be chosen.For a givenmodel the integrals in eqn ( 5 ) may be solved numerically. Tt would be hopeless totry this with one of the common methods of integration? for instance, with the rule ofSimpson. Such multiple integrals can be solved numerically by the following MonteCarlo method. Consider the integral J = Jf(xl,x2, . . .pN) dx, . . . dxN. 4, therange of integration, may be the interval 0 -= x1 <2n, 0 < x2 <2nY . . ., 0 e x , < 2n. Aset of random numbers r1,r2, . . ., r, each placed between 0 and 271 is now generatedand f(rl,r2 . . . rN) is calculated. Then (27~)~f(r~, . . . r,) is a first approximation ofthe integral. The procedure is repeated as often as possible and the average is taken.454 RADIUS OF GYRATION OF STIFF CHAIN MOLECULESIt converges towards the correct value of the integral. From the fluctuations of thesingle results one can estimate the uncertainty of the calculation.If eqn (5) is appliedto the model of thepolymer molecule, thevariables are the angles of rotation, azimuthand latitude of the k s t bond of each chain and the coordinates of the centre of mass ofthe second of the two chains.In the following way the calculation programme was worked. A random con-formation was constructed and was put with its centre of mass into the origin of thecoordinate system.U(i) = 21 . & (6)zf is the number of intramolecular contacts, i.e., the number of distances h,, betweentwo potential centres j and k with x1 <hjk<x2.Distances smaller than x1 are notallowed because this would mean U(i) = 00. As soon as such intramolecularoverlapping took place, the conformation was rejected and the construction of thechain was started with new random numbers. Because of the stiffness of the chainintramolecular overlappings did not happen often. At N = 300 with a probability ofabout 0.8, no ring closure occurs during the construction of a conformation.A second random coil with a random starting direction relative to the first wasconstructed and also put with its centre of mass into the origin of the coordinatesystem. Then it was shifted in small steps along the axes of the coordinate systemwithin the total range in which contacts between the two coils could occur.Thewidth Ad of the steps was about as big as the thickness of the chain. To each positionof the two coils relative to each other the intermolecular potential energy U(1,2) wascalculated. It is infinity if two centres of potential have a distance smaller than x1otherwiseHere z1,2 is the number of intermolecular contacts (x, < hlj2k < x2).With the knowledge of U(l),U(2) and U(1,2) the integrands in eqn (5) can be cal-culated for a given pair of random conformations in a specified position relative toeach other. For the integration with respect to the coordinates of the centre of masss2 in eqn (5) in each position of the second coil the integrand is multiplied with theproper volume element, which is 1/6 of a spherical shell with the thickness Ad and theradius d = distance between the centres of mass of the two coils.Together with A2 the mean-square-radius of gyration was calculated.In theaveraging r2 of any random conformation has to be multiplied with Fl(i) as a weightfactor. According to a2 = (r2)/<r:) the expansion coefficient CI is calculated. Thesubscript 0 corresponds to the unperturbed coil, i.e. to a solution in which A2 = 0.Radii of gyration and expansion coefficients of model-chains have been calculatedwith the Monte Carlo method already by Wall et aL9-15 In contrast to the presentpaper, Wall uses the sites of a lattice for the construction of his model chains.U(i) was calculated byU(1,2) = 21.2 . & (7)RESULTSVIRIAL COEFFICIENTSFrom eqn (5)-(7), A2 is obtained as a function of kT,k Parts of these A,-valuesare shown in fig.4. A2 becomes zero at about kT/e = -2.5. If &'is held constant,then from A2 = 0 a certain temperature T = 8 follows. If the temperature is fixedon the other hand, one obtains a certain depth of potential E~ at which A2 vanishes.For this discussion it is convenient to define the parameteR. G . KIRSTE 55- 6 -8 - 10kT/s00’ I,’ ,‘0 . ,, .’ I. . ,4 2FIG. 4.--A2 from Monte Carlo calculations against kT/e. N is the number of potential centres fromfig. 2. For comparison with CTN M = 980 N. 0 Q 0 = results from the calculation.Fro. 5.-A2 against T, results from the calculation56 RADIUS OF GYRATION OF STIFF CHAIN MOLECULESWith E = const, it follows z = 1 - O/T.At the &point, z = 0 ; in the athermicsolution z = 1, in the endothermic z < 1, and in the exothermic z> 1.In fig. 5 the calculated A,-values are plotted against z. Parameter is the chainlength. Fig. 4 and S give a rough impression of the accuracy of the calculation. Infig. 6 the numerically calculated A,-values are drawn against A4 together with experi-mental data?. The results are also compared with the formula of Casassa 7 *A2 = (N/2)(fi/Mi)h(z), with z = (4n)-3(P/Mi)(M*/Kt), (9)KO = (r$)/M, h(z) = [l - exp (- 5 . 6 8 ~ ~ ; 3j]/5.68za,3, and a; -a% = 2.043~.For KO the experimental value was used. With eqn (9) Az is obtained as a function ofM and fl/Mi.isthe excluded volume of a spherical segment. u(x) is the potential for the approachingof two segments.= 4nJ[1 -exp ( -u(x)/kT)]x2 dx (10)Mo is the molecular weight of a segment.FIG.6.-& against Mfor CTN and the model. Full lines, numerical calculation of the present paper(parameter 7) ; broken lines, theory of Casassa, eqn (9) (parameter PIM:). 0, measured values ofCI" (13.9 % nitrogen) in acetone ' S *One can see in fig. 6 that at high molecular weights the numerically calculatedcurves become more and more parallel to the curves of Casassa. But the distancebetween the curves is different indicating that eqn (9) is not applicable to our model or/3 is not proportional to 'c.In fig. 6 a discrepancy between experiment and the Monte Carlo model calculationoccurs if the heat of dilution is taken into account.The experimental values lyingnear the athermic curve ('c = 1) are obtained from exothermic solutions. Thecalculated curves can be shifted downwards if the thickness of the model chains islowered. With 10.2 A (= 0.61, I = 17 A) the thickness of the CTN-chains indeedmight have been overestimated. In an additional calculation, substituting, x1 = 8 Aand xz = 12 %L for N = 30, A,-values were obtained which are smaller by a factor ofabout 0.7 than those plotted in fig. 6R. G . KIRSTE 57RADII OF GYRATION AND EXPANSION COEFFICIENTSIn fig. 7 the radii of gyration obtained from the model calculation are plottedagainst N in a logarithmic scale with z as a parameter. The asymptote agrees fairlywell with the function (r:) = bL/6 (b = 20, L = N=(bL/6)O*’ = 1.83 No*’, theasymptote in fig.7 is 1.74 NO.9. For smaller N the calculated function shows theTABLE EXPANSION COEFFICIENTS a OF THE MODEL CHAIN AND OF CTN IN ACETONEmodel calculation CTN in acetone 1.N1030100300900aM T = 0.5 t = l r = 29 800 1 1 129 400 1.000 1 1.0002 1.o00398 000 1.003 1.005 1.007294 000 1.007 1.012 1.017882 000 1.03 1.04 1.055M 0:82 OOO 1.005150 000 1.016320 000 1.04740 000 1.051 660000 1.073 650 000 1.13transition behaviour between coil and rod. The main result of fig. 7 is, that the sec-ond virial coefficient has almost no influence on the radius of gyration. In table 1the numerically calculated expansion coefficients are listed together with experimentaldata. In agreement with fig.7 all the expansion coefficients are approximately equalto unity.ndE WLtocloo 10’ lo2 lo3 lo4NFIG. 7 . T h e radius of gyration as a function of chain length and the polymer-solvent interaction ;0 and 0, results from the calculation. For comparison with CTN : unit of length is I =17 A,L = 17Nand M = 980N.DISCUSSION OF ERRORSMonte Carlo calculations with respect to errors are to be treated like experiments.In the calculation of the standard deviation of A2 and (r2> one has to account for thedifferent statistical weight G of the trials. It isan58 RADIUS OF GYRATION OF STIFF CHAIN MOLECULES4The statistical weight of A2 and <r2> obtained from a group of q trials is thenandof trials. The standard deviations of these averages of groups were calculated byGi(A2)Gt(r2) respectively.The A2 and <r2)-values in fig. 4-7 are averages of groups4 i= 1i = 10 5 8a(fm) = (i i= 1 ~ i ~ f ; ) /C i= 1 ~ i * (13)In this way the standard deviations in table 2 were obtained. fmeans A2 or r2, fm isthe average from all groups,f' is the difference between this total average and theTABLE 2.-NUMBER OF MONTE CARL0 TRIALS AND STANDARD DEVIATIONS OF AND (r2>N number of trials*- 13 97010 986 * This is the number of trials for the30 1 058 determination of <r2>. Thenumberof pairs of threads for the determin-ation of Az is half as large. 100 422300 334900 914standard deviations for N = 9004 . 4 2 ) x 104 (cm3g-2)kT/& = -2.4 1.13 1.27 = 2.5 %kT/E = -5 0.24 0.38 = 0.75 %kT/& = 00 0.21 0.39 = 0.75 %kT/E = 2 1.16 0.39 = 0.75 %a(<rW.s) (unit : I )m 2 1 - ' lo5 2 4 6 106 2 4 6 10' 2 4 6MFIG.&-Radii of gyration of DNA against M. r is the root mean square and the z-average, M is theweight-average. Light-scattering measurements : 0, Doty (1 958)' ; , Cohen and Eisenberg(1966)" ; 0, Zimm et al. (1968)18 ; 0, new measurements on trout sperm DNA in 0.05 m trinatriumcitrate and 0.15 m NaC1.l9 Full line = eqn (14)R. G . KIRSTE 59average of the trials of a single group i, Gi is the statistical weight of a single group ands is the number of groups. Eqn (13) becomes invalid if the number of groups is toosmall or if the weight of few groups dominates too much.THE RADIUS OF GYRATION OF DESOXYRIBONUCLEIC ACIDIn fig.8 the root-mean-square radius of gyration of DNA in buffered aqueoussolution is shown as a function of M.16-19 The solutions have a positive secondosmotic virial coefficient. A straight line passing through the experimental datawould have a slope much greater than 0.5, but this is not a consequence of polymer-solvent interaction. The DNA-chain is so stiff that the samples used for the measure-ments lie in the transition range between rod and coil. A formula for the relationbetween t and M in this transition range can be derived using the persistent chainmodel of Kratky and Porod 2 o s 21 :a is the length of persistence and L the length of the extended chain. The full line infig. 8 was calculated with a = 1 100 and L = (M/200) A.G. V. Schulz and E. Penzel, Makromol. Chem., 1968, 112, 260.E. Penzel and G. V. Schulz, Makromoi'. Chem., 1968, 113,64.H. Eyring, Phys. Rev., 1932,39,746.H. Baumann, PoZymer Letfers, 1965, 3, 1069.R. G. Kirste, Int. Symp. MacromoZ. Chem. (Tokyo-Kyoto, 1966), vi, 163.B. H. Zimm, J. Chem. Phys., 1946,14, 164.E. F. Casassa, J. Chem. Phys., 1959,31,800.F. T. Wall, L. A. Hiller Jr. and D. J. Wheeler, J . Chem. Phys., 1954, 22, 1036.lo F. T. Wall, L. A. Hiller Jr. and W. F. Atchison, J. Chem. Phys., 1955,23,913 ; 1955,23,2314 ;and 1957, 26, 1742.l1 F. T. Wall and J. J. Erpenbeck, J . Chem. Phys., 1959,30,634; 1959,30, 637.lZ F. T. Wall, S. Windwer and P. J. Gans, J. Chem. Phys., 1962,37,1461; 1963,38,2220 and 2228.l3 P. J. Gans, J. Chem. Phys., 1965,42,4159.l4 S . Windwer, J. Chem. Phys., 1965, 43, 115.l6 P. Doty, Proc. Nat. Acad. Sci., 1958, 44,432.l7 G. Cohen and H. Eisenberg, Biopolymers, 1966, 4,429.l9 R. G. Kirste and B. E. Zierenberg, New Measurements (Mainz, 1969).'O 0. Kratky and G. Porod, Rec. Trav. Chim., 1949, 68, 1106.21 H. Benoit and P. Doty, J. Phys. Chem., 1953, 57,958.' E. F. Casassa and H. Markovitz, J. Chem. Phys., 1958,29,493.P. Mark and S. Windwer, J. Chem. Phys., 1967,47,708.J. A. Harpst, A. I. Krasna and B. H. Zimm, Biopoi'ymers, 1968, 6, 595