Random Flight Model in the Theory of the Second Virial Coefficient ofPolymer SolutionsAndreas C. Albrecht Citation: J. Chem. Phys. 27, 1002 (1957); doi: 10.1063/1.1743923 View online: http://dx.doi.org/10.1063/1.1743923 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v27/i5 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 27. NUMBER 5 NOVEMBER. 1957

Random Flight Model in the Theory of the Second Virial Coefficient of Polymer Solutions*

ANDREAS c. ALBRECHTt

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Received June 12, 1957)

The theory of the second virial coefficient, A2(M2,T), for dilute polymer solutions is investigated from the point of view, first developed by Zimm, of the Ursell-Mayer-like development in terms which successively represent increasing numbers of contacts between molecular subunits of flexible chain molecules. The methods used by Fixman in treating excluded volume problems in the random flight model are here employed. Evaluation is carried out to the triple contact approximation. A formal analysis of all higher terms is pre­sented and the contribution of one class of terms is evaluated exactly. The remaining terms appear to be inaccessible to numerical evaluation by ordinary methods so that a rigorous discussion of the asymptotic behavior of A2 is precluded. Semiquantitative arguments, however, indicate favorable agreement with experiment, provided intramolecular excluded volume effects are considered.

I. INTRODUCTION

T HERE have been numerous attempts to explain theoretically the nonideal behavior of dilute high

polymer solutions. A brief listing of several theories may be found in a recent paper by Isihara and Koyama l and will not be recounted here.

It can be stated that the object of any theoretical discussion of the problem is to predict correctly the nature of the virial coefficients which appear in the ex­pansion of the osmotic pressure in terms of concen­tration. For a two-component system this reads

(1)

where M2 is the molecular weight of the solute, and c is the concentration expressed in grams per unit volume and the A/s are the virial coefficients. In particular for dilute solutions it is the prediction of A 2(T,M2) that is of interest.

The general theory of nonelectrolyte solutions of McMillan and Mayer2 (see also Kirkwood and Buff2) leads to expressions for the virial coefficients in terms of integrals over certain continuous distribution functions. The second virial coefficient is written in the notation of McMillan and Mayer as

(2)

where F2(1,2), F I (1), and F1(2) are special cases of the general distribution function FN(1,2,· . ·N) defined as follows:

FN(1,2,· . ·N)d(1,2,· . ·N)/VN (3)

is the probability (strictly, for an infinite system) that the N solute molecules 1, 2, ... , N lie in a region of

* This work was supported by the Office of Ordnance Research. t Present address: Department of Chemistry, Cornell Uni­

versity, Ithaca, New York. 1 A. Isihara and R. Koyama, J. Chern. Phys. 25, 712 (1956). 2 W. G. McMillan and J. E. Mayer, J. Chern. Phys. 13, 276

(1945); J. G. Kirkwood and F. P. Buff, J. Chern. Phys. 19, 774 (1951).

generalized configuration space between (1,2,· . ·N) and (1,2,·· ·,N)+d(1,2, .. ·N), where (1,2,·· ·,N) symbol­izes a specific set of all coordinates internal and external of the N molecules and d(1,2, .. ·N) is the appropriate element of hypervolume. V is the volume of the system and No is Avogadro's number. These functions are normalized (strictly true for V---too) so that

V-N f FN(1,2,·· ·N)d(1,2,·· ·N)=1. (4)

Now all theories of A2 which employ continuous dis­tribution functions must be derivable from Eq. (2). They differ only in their choice of FI and F2 as dictated by the model used.

In 1946, Zimm3 presented a development of A2 based on Eq. (2) which lays the basis for a detailed analysis of the interaction of two flexible high polymer molecules. That is, it forms the framework for any theory which demands the explicit dependence of F I and F 2 on internal as well as external coordinates. The treatment takes the form of an expansion in terms which successively repre­sent a greater and greater number of simultaneous con­tacts between molecular subunits of each of two mole­cules. It was applied to the random flight chain model, where numerical evaluation was carried out to the second contact term. While this was sufficient to indi­cate the molecular weight dependence of A2 for very weak effective interactions, it was clear that a more comprehensive understanding of molecular weight de­pendence would have to await the evaluation of the higher contact terms-an apparently very difficult task.

Simpler models which lead to complete analysis have been treated. Flory4 used a penetrable uniform density sphere (F) which was later modified to a penetrable Gaussian model by Flory and Krigbaum· (FK) which in turn was modified by Isihara and Koyama l to the more

3 B. H. Zimm, J. Chern. Phys. 14, 164 (1946). 4 P. J. Flory, J. Chern. Phys. 13,453 (1945). 5 (a) P. J. Flory, J. Chern. Phys. 17, 1347 (1949) ; (b) P. J. Flory

and W. R. Krigbaum, ibid. 18, 1086 (1950); (c) See also, T. B. Grimley, Proc. Roy. Soc. (London) A212, 339 (1952).

1002

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SEC 0 N D V I R I A L C 0 E F F I C lEN T 0 F POL Y MER SOL UTI 0 N S 1003

complicated "summed" Gaussian density model (IK) which correctly represents the averaged or "smoothed" density of a random flight chain. Common to each of these models is their requirement of explicit dependence of FI and F2 on external coordinates only. Physically speaking, detailed interaction of flexible chains is not treated; the instantaneous configurations are taken to be smoothed to an average spherically symmetric distri­bution before the interaction of two molecules is con­sidered. The FK and IK theories, which are basically very similar, together with a recent semiempirical refinement of the FK theory by Orofino and Flory,6 have proved fairly successful in their correlation with experi­ment. From the point of view of the interaction of two flexible high polymer molecules, the theories involve approximations so drastic7 that their relative degree of success would appear somewhat surprising. In general, one should expect that model to be the more realistic which explicitly takes into account the greater number of coordinates of the bimolecular systems.

As has been suggested,S one might attempt to refine the smoothed density models by introducing fluctua­tions. Alternatively, by employing the 0 function as introduced by Fixman9,lo in both the intra- and inter­molecular excluded volume problems, one is tempted to explore further the nature of the higher terms of the general development presented by Zimm. The latter course is the one taken in the present investigation.

After a presentation of the general development, the random flight model is formally treated. Quantitative evaluation through the triple contact term is carried out, and all higher contact terms are analyzed into two classes, one of which is evaluated exactly, while the other is discussed in semiquantitative terms. The ap­parent great difficulty in obtaining a quantitative esti­mate of the latter class precludes a rigorous discussion of the asymptotic problem (M2--+oo) based on this model. The results applicable for the domain of weak subunit­subunit interaction (poor solvent) are discussed. The importance of the effect of intramolecular excluded volume on A2 in the domain of stronger interactions is emphasized and critically treated.

II. THE GENERAL DEVELOPMENT OF A2

The general theory of Zimm3 as modified by using the mathematical procedure of Fixman9,lo is now presented. Important points of the development are discussed.

6 T. A. Orofino and P. J. Flory, J. Chern. Phys. 26,1067 (1957). The author wishes to thank Professor Flory for the opportunity of seeing the manuscript prior to publication.

7 Common to the smoothed density models is the factorization of the internal distribution junction, F.(l)l, into a product of single molecular subunit (or segment) distribution functions. The segment distribution functions are taken to be identical for all segments in the (FK) model; while in the IK model they are con­sidered to be dependent on the segment position in the chain [see, for example, Eqs. (2.10) and (2.12) of reference 1].

8 Zimm, Stockmayer, and Fixman, J. Chern. Phys. 21, 1716 (1953).

9 M. Fixman, J. Chern. Phys. 23, 1656 (1955). 10 M. Fixman, thesis, Massachusetts Institute of Technology

(1953).

The distribution functions required according to Eq. (2) are defined as follows:

F2(1,2)=exp[ -U(1,2)/kT], (Sa)

FI(l) =exp[ - U(l)/kT], (Sb)

where U(l) and U(l,2) are special cases of U(1,2,· .. ,N) the potential of average force on N molecules.ll Pairwise additivity of the potentials of average force between the molecular subunits is assumed (superposition principle) so that one may write, assuming each molecule to con­tain n subunits,

n n

U(1,2) = U(1)+U(2)+ L L U(r.x) (6) .~I X~l

or

F2(1,2)=FI(1)Fl(2) exp[ - L L U(r.x)/kT], (7) • x

where the Greek letters are dummy indices, always appearing in pairs, referring, alphabetically to a subunit of molecule 1 and a subunit of molecule 2, respectively. The r.x are vectors connecting the denoted subunits.

A new function is introduced in exact analogy to the Ursell-Mayer treatment in imperfect gas theory:

1+x.x=exp[ -U(r.x)/kT]

so that we write using (7) and (8) in (2)

A 2= (N o/2VM22)

(8)

X f FI(l)FI(2)[1-l! \I(1+x.x)]d(1,2). (9)

A second important assumption is now introduced (the first being the use of superposition). It is assumed that the function x.x is of such a short-range nature that it goes from its peak value (at r.x=O) to zero with a variation in r.x sufficiently small so that the product FI(1)FI(2) remains unchanged. Thus when integrating over the relative positions of any two subunits the product F I(l)FI(2) may be removed from the integrand and taken to be equal to its appropriate value (con­stant) obtaining at any r.x within the range where x.x does not vanish. The integral remaining over x.x is ultimately considered an empirical parameter which shall be written as

x.x= - f x.xd(r.x). (10)

It is a function of temperature and dependent on the environment of and nature of the molecular subunits. It may be considered as an effective volume excluded by one molecular subunit to another. The explicit functional nature of x(r.x) is entirely immaterial to the argument

11 Given an assembly of M molecules (M>N), U(1,2,·· ·,N) is the potential whose negative gradient (generalized) in the sub­space specifying (1,2,'" ,N) is the average force exerted on the configuration (1,2,'· ·,N) by all other molecules, M-N, of the assembly.

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1004 ANDREAS C. ALBRECHT

as long as the assumption regarding its short-range nature can be reasonably made. It is seen that this manipulation of the integration can most conveniently be achieved mathematically by substituting into (9)

(11)

where o.x is a three-dimensional Dirac delta function peaked at r.x=O. The substitution (11) is purely a mathematical procedure valid under the above assump­tion and in no way represents any new assumption about the functional nature of x.x (and hence of V(r.x».

Substituting (11) in (9) and expanding the double product, one finds

A 2= (No/2VMi)fF1(1)Fl(2)[L L X.xo,x • x

- L L L L X.xX!'.o.xo!,.+· .. Jd(I,2), (12) I( x p. v

with K~J.I, and X<v when K=J.I. For convenience each term of this expansion shall be referred to as Term and the individual terms arising from the multiple summa­tions shall be referred to simply as terms, without capitalizing. Thus altogether there are n2 Terms, the kth

Term having (~2) terms when the multiple summation

is expanded. Performing the integration one finds for a given term

of the kth Term

k pairs

= VPk-l(O!'.,.· ',Opu," ·).X II X.x. (13)

The integration is carried out as follows:

(1) X.}.. is removed from the integrand and integra­tion is carried out over coordinates specifying the rela­tive positions of subunit K of molecule 1 and subunit X of molecule 2. These subunits are arbitrarily chosen to take part in the first of k contacts. This integration yields a new distribution function which we write as F 1(1·2) to symbolize the fact that the presence of 0 • .,. in the integrand has figuratively speaking brought molecules 1 and 2 into contact forming a single bimolecular cluster whose configuration is symbolized by (1·2).

(2) Integration is carried out over the coordinates which locate the bimolecular cluster with respect to an external origin to give VF1(1·2) [. The quantity F1(1·2)[ is the distribution function which describes the internal configuration only of the bimolecular cluster.

(3) Finally integration is carried out over all internal coordinates to give P k-l(O.}..,· .. ,Opu,' .. ).}.. defined by

(14)

which may best be thought of as a conditional proba­bility that given an initial contact between subunits K

and X there exist also contacts between subunits J.I and v, "', p and u, "', and so on, for a total of (k-l) addi­tional specified contacts.

We now have with (12) and (13)

A 2= (No/2M22)

X[L X.X - L X. XX!'.P1(O!'.).x+··· J, (15) «,). IC,A.J.&,V

where detailed conditions on the multiple summations have been supressed.

To this point the development is quite general (for a monodisperse polymer solution). Use has been made of the superposition principle (perhaps the most serious approximation) and x.x is assumed to be short-ranged.12

Without much loss of generality we assume that all subunits are identical. Thus, dropping the subscripts on X .x, one obtains

A 2= (Non2X/2M22)

X[1-Xn-2 L P 1(O!'.).x+·· 'J, (16) It,).,,,,. "

or, finally,

A2(T,M2) = (NoX/2Mo2)H(X,M2)' (17)

where Mo is the molecular weight of a molecular subunit and the function H(X,M2) represents the expression in the brackets of (16) and is the function, dependent on model, which gives the molecular weight dependence of A 2• For vanishing X, H goes to unity and one obtains a result known as the single contact approximation for A2 which is independent of model (unless theoretical evalu­ation of X is to be considered).

The random flight model is now introduced and analytic expressions for the conditional probabilities P k- 1 [Eq. (14)J are obtained.

III. RANDOM FLIGHT MODEL

(1) Determination of H(X,M 2)

The random flight model is taken9,10 as a sequence of n vectors, ri, (i=I, 2, "', n), which represent segments (the molecular subunits) of the chain. The length of a given segment shall be Gaussianly distributed about a mean square value of b2• The probability that molecule 1 have the internal configuration specified by the n vectors r i within the range r i to r i+dr i is written as

.. F1(1)rd(l)r= II (3/21rb2)i exp( -3rl/2b2)drj (18)

i=1

12 The product of the single molecule distribution functions, F,(1)F(2), is not a function of the relative positions in V occupied by molecules 1 and 2. Thus as far as the single contact approxima­tion is concerned this assumption is unnecessary. Detailed con­sideration shows that this assumption becomes increasingly necessary as higher Terms are considered and becomes severely tested only when a large numher of simultaneous contacts are important.

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SEC 0 N D V I R I A L C 0 E F F I C lEN T 0 F POL Y MER SOL UTI 0 N S 1005

and the probability that a bimolecular cluster in contact at a given point have a given internal configuration within the appropriate range is

2n

F2(1· 2)rd(1' 2)r= II (3j27rlr)t exp( -3r !/2b2)dr;, (19) i=l

where the product runs over segment vectors of both molecules the distinction not being explicitly stated in the notation.

Using9 the theorem of Wang and Uhlenbeck13 on multivariate Gaussian distributions, we obtain

where C. is formally defined in Appendix A and we shall henceforth write s for k-l. To be consistent with previ­ously used8,9 notation, we put

(21)

then substitute (20) into the bracketed expansion of (16) to obtain

H=h(z)=[1-zn-1 L c1-t /(,)..,1',11

/(,X,jJ, v, p,er

The problem that now remains is to evaluate the indi­cated multiple summations. First, however, the de­terminants C. shall be examined in detail. This shall lead to a general classification of terms into two groups which will greatly aid in the discussion that is to follow.

(2) Discussion and Analysis of C.

C. is the determinant belonging to the kth Term of (22). This is the Term for which molecules 1 and 2 can be considered in simultaneous contact at k (or s+ 1) points. Choosing one point of contact as a reference point it is possible to associate with each of the additional s con­tacts a loop containing sections of both molecules. The loop to be associated with the ith additional contact, for example, begins at the reference point, includes a section of molecule 1 up to the contact point of the ith additional contact and closes by including the section of molecule 2 which leads back to the reference contact. The diagonal element, C;i, of C., as defined in Appendix A, represents the number of segments contained in the loop just described. That is C ii is the size of the ith loop. Similarly C jj is the size of the loop to be associated with the jth additional contact, and so on for the s diagonal elements of C •. The s loops so defined obviously contain certain segments in common. The number of segments contained commonly by loops i and j is

13 Ming Chen Wang and G. E. Uhlenbeck, Revs. Modern Phys. 17,323 (1945).

b t---------j 2 I, "

a --------- I

A (12) B (21)

FIG. 1. Diagrams for double contact types. The vertical solid lines in a given diagram, from left to right, represent molecules 1 and 2, respectively. The dotted lines connect the contact segments, which are labeled alphabetically in molecule 1 and numerically in molecule 2. The parameter x, represents the number of segments which separate contact segments a and b, while YI represents the number of segments separating contact segments 1 and 2. In parentheses is given the permutation which corresponds to the given diagram. It is obtained by writing the contact segments of molecule 2 proceeding from bottom to top.

represented by the element C ij in c., etc., for all off­diagonal elements. Clearly Cij=Cji•

Now the detailed conditions on the multiple summa­tions appearing in (22) indicate that while the order of contact segments on molecule 1 is preserved (i.e., for k=3, K~P.~p) the contact segments of molecule 2 (for k=3-segments A, v, and u) are allowed to vary inde­pendently. This gives rise to a set of types of multiple contacts, each type corresponding to one of the k! (for k contacts) possible orderings of the contact segments of molecule 2. This division into types is necessary, for in general the C., determined as just described, are not the same for all types. Physically these types correspond to varying degrees of looping back of molecule 2 in achieving contact with molecule l.

The discussion is greatly aided by using a diagram­matic representation of the various types of multiple con tacts. In Fig. 1 are depicted the two (2!) possible types which can appear in the double contact Term and in Fig. 2 the six (3!) types which appear in the triple contact Term. The vertical lines in a given diagram refer to the two molecules, respectively, and the dotted lines connect those segments involved in contact. The contact segments of molecule 1 are labeled alphabetically be­ginning from the lower end (always preserving the order) while the contact segments of molecule 2 are labeled numerically according to the alphabetical posi­tion of their partner in molecule 1 (al, b2, c3, etc.). This temporary change of notation (instead of using KA, p.v, pu, etc.) lends itself especially well to a short-hand notation for types to be presently discussed. The parameters Xi and Yi represent the number of segments separating the contact segments of molecule 1 and of molecule 2, respectively. Land S are simply L Xi and L Yi, respectively, and will be used in Appendix C.

As suggested in Figs. land 2 (in parentheses), a con­venient shorthand notation for labeling the different types is achieved by listing the numbered contact segments of molecule 2 in order as they appear proceed­ing from the lower end to the upper. In general to generate all possible types of say, the k contacts one need only write all permutations of the ordered number set 1, 2, "', k, and interpret each permutation in the

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1006 ANDREAS C. ALBRECHT

tc ~"-------j 3~ L :~ -------- 2: 5 a -------- I

A (123) a (I 32)

C (213)

b --;>(-- 2 x~ t" "i I J. -a. ",' '..... s'·

E (312) F (321)

FIG. 2. Diagrams for triple contact types. The notation is similar to that of Fig. 1. The four parameters Xl, X2, Yl, Y2 specify the separation, in terms of numbers of segments, of the contact segments of molecules 1 and 2, respectively. L specifies the separa­tion of the "first" and "last" contact segment of molecule 1 and S represents this number for molecule 2 (~x;=L, ~ y;=S). In pa­rentheses is given the permutation which corresponds to the diagram.

diagrammatic sense. Since there are k! types belonging

to the kth Term which has (~2) terms there must be

(~2) / k! terms per type.

Consider now, diagram A of Fig. 2. Choosing al as the reference contact we write C. as just outlined in terms of the two loop sizes (Xl+Yl), (Xl+X2+Y2+Yl) and their overlap, (Xl+Yl). Thus

C2= I (Xl+Yl)

(Xl+Yl)

(Xl+Yl) I (Xl+X2+Y2+Yl)

(23)

Upon simple transformation a diagonal determinant is obtained with elements (Xl+Yl) and (X2+Y2). The formal definition of C. is, as expected, not unique, since determinants are invariant to a variety of transforma­tions. In fact, every equivalent representation of C. corresponds to a particular manner in which s loops are chosen. In particular, in Fig. 2A we can choose loop 1 to be Xl+Yl, and loop 2 to be X2+Y2. These contain no segments in common and we are led to a diagonal C2

instead of (23). This suggests that an alternate method of defining

loops be adopted in order to lead directly to a simplified C 8. If the s loops are chosen in a manner such that there is a minimum number oj loops which overlap (contain common segments) the desired simplification of C. is

obtained. This corresponds to transforming the determi­nant as originally defined by applying successive ele­mentary transformations of the kind which subtracts row i from row j and column j from column i until a determinant with a minimum number of diagonal terms has been obtained. Determinants are invariant to such transforma tions.

This method of choosing loops leads directly to two classes of contact types: Class I, containing those types for which C. is diagonal (these shall be referred to as "no overlap" types) ; and Class II, containing all those types (the remainder) for which C. cannot be diagonalized (these shall be referred to as the "overlap" types). It can be shown that there are only two types which belong to Class I for a given Term. These are the types whose representative permutation is either the perfectly or­dered number set or the perfectly reverse ordered num­ber set. The remaining types (k!-2 for the kth Term) belong to Class II. The Class I representatives in Fig. 1 are diagrams A and B, and in Fig. 2 are diagrams A and F. Numerical evaluation of the contribution of the Class I types is possible. However, it appears extremely difficult to evaluate exactly the contributions of Class II, although it will be shown that type for type the contri­bution of the latter must be smaller than the contribu­tion of Class I. The Class II types in the third Term have been evaluated, but only as a result of laborious calculations.

There are two kinds of symmetry which appear among the types. The first symmetry is a property common to both classes; the second belongs only to Class II. The first is a symmetry which causes two different types to yield two C. differing only in that in one the Yi are labeled in just the reverse manner-these lead to equivalent contributions once the sum has been carried out. This is a symmetry inherent in the process of two molecules (linear) approaching each other where they may be oriented either head to head or head to toe. This corresponds to the two permutations 1, 2, ... , k and k, k-l, ... , 1 for Class I, and similarly (for Class II) to all other pairs of representative permutations achieved by analogous transpositions on the ordered and reverse ordered sets. Thus each of the following pairs of dia­grams in Figs. 1 and 2 have equivalent contributions: lA, 1B; 2A, 2F; 2C, 2E; and 2B, 2D.

The second symmetry can be seen as follows. Con­sider a plane normal to and bisecting the two solid lines of a given diagram. It can make no physical difference whether the overlap occurs on one side or at its mirror image through this plane. Types so related must lead to the same numerical contribution. In Fig. 2, therefore, the contribution of Type B is equal to that of C and the contribution of Type D is equal to that of Type E. The number of unique C. that require evaluation in the sense of performing the multiple summation is not k! but (k!+2)/4. For k=3, therefore, there are only two C. to evaluate, not six, the symmetry factors being appro­priately applied to give the correct final answer.

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SEC 0 N D V I R I ALe 0 E F FIe lEN T 0 F POL Y MER SOL UTI 0 N S 1007

With the discussion of C. completed we turn first to the evaluation of h(z) through the third Term, then finally to a semiquantitative discussion of the higher Terms, where further use shall be made of the analysis just presented.

(3) Evaluation of h(z) through the Third Term

Unless machine methods can be applied14 we are forced to replace the summations in Eq. (22) by integrals. For sufficiently large n this introduces negli­gible error. Where proper counting [as dictated by the expansion of the double product in Eq. (9)J gives

(~) terms in the kth Term, integration over appro­

priate differential elements with lower limits zero and other fine points of the restrictions in the multiple summation glossed over counts an effective n2k jk! terms.

Even for the last Term, containing (:2) terms, the

integration counts correctly if n is large with respect to unity. In fact, in permitting the lower limit to be zero one is including singularities in c.-! which correspond physically to zero loop sizes. The integrations, however, do not diverge and in fact give the correct evaluation of the multiple sum for n!»1.

There are compelling reasons for actually preferring n always to be very large. The random flight Eq. (18) is itself valid only for large n. Furthermore, we have no unigue way to specify the size of a segment and hence to fix a physically significant finite value of n. By con­sistently using integration for summation, we arrive at final results containing only parameters [M2, n2X, nb2; see Eg. (26) below J which are invariant to changes of segment size. Figuratively, the "pearl necklace" is transmuted into spaghetti.

Taking account of the symmetry conditions discussed above, integration is to be carried out over the minimum number of determinants (C.-!) necessary to account for all types present. For a given type-say (132)-the limits on the integration are easily set as n~ c~ b~ a~ 0 and n ~ 1 ~ 3 ~ 2 ~ 0 (where the numbers of course are to be taken as symbols). The letters are always in alpha­betical order while any crossing over of the numbers leads to a new permutation or another type. Since the general determinant, c., is a function only of the rela­tive positions of the contact segments it was found con­venient to integrate first over the positions of the first contact segments (those appearing at the lower end of a given diagram). Using reduced variables,!5 this integra-

14 The summations for the third term were programmed for the IBM 650 Computer. It was found that for n as small as 10, the time required for the summation was already prohibitive (about four hours) and essentially doubled for each additional increase of n by unity. Convergence to the value obtained by integration ap­peared to be very slow. Instructions in programming and aid in using the machine provided by the Cornell Computing Center are gratefully acknowledged.

aWe write L'=L/n, S'=S/n, and x/=x./n, Y;'=Yi/n where the unprimed variables are as used in Figs. 1 and 2. Here the primes are immediately suppressed.

tion introduces a factor (1- L) (1-S) into every integral. For the double contact Term we have for the multiple

sums the two Class I types (symmetric)

and for the triple contact Term we have for the two Class I types (symmetric)

2n3flfl fL.f S (1- L) (l-S) (L+S- Xl- Yl)-! o 0 0 0

X (Xl+Yl)-JdLdSdx1dYl (2Sa)

and for the four Class II types (also symmetric)

X (Xl+Yl)+XlYl]-!dLdSdx1dYl. (2Sb)

These integrals are evaluated (see Appendix B) to give finally, with (22) and (17),

A 2= (NoXj2Mo2)[1-2.86Sz+9.726zL ... ]. (26)

The double contact Term is in agreement with that obtained by Zimm3 (where the symmetry factor of 2 was erroneously omitted) and Fixman.lO It is interesting to point out that this is an expansion in powers of a parameter z which is very nearly a ratio of the single contact excluded volume (intermolecular) of two rods of n segments (n2X) to the excluded volume of a rigid sphere of radius (nb2j6)!, the rms. radius of a random flight coil [see Eg. (21)].

A discussion of Eg. (26) in comparison with the smoothed density models will be given. Next, however, a semiquantitative analysis of the higher Terms IS

presented.

(4) Discussion of Higher Terms

(a) Class I Types

The determinant for the general no-overlap term of the kth Term being diagonal, reads simply

• C/ = II (Xi+Yi), (27)

i=l

where the superscript denotes no-overlap (Class I) and, as before, s= k-1. The multiple integration using the same method already outlined, excluding the symmetry factor 2, is written

T k = f ... J (l-L)(l-S)

0< 2: x, =L< 1 O~2:y,=S~l .

xII {(x,+Yi)-Jdxidy;}dLdS (28) i=l

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1008 ANDREAS C. ALBRECHT

and the integration is performed as described in Ap­pendix C to give

Tk=---------------Ck+ 1) Ck+3)rCk/2)

X [Ck2+3)I k-l- (k-l)2-(k-l)/2], (29) where

1 .. /4

I k-l= 0 sink-lf)d8~2-k(k+I)/2(k+2)/k. (30)

The total contribution of Class I types to the kth Term is just 2T k.

Now an argument shall be presented which indicates tha t none of the k! - 2 Class II types can alone equal T k

in the value of its contribution. If we let the true value for the kth Term be symbolized by B k it will then be possible to write

(31)

The lower limit assumes that Class II contributes nothing, the upper assumes that, type for type, their contribution is equal to the Class I type, T k.

(b) Class II Types

We recall that the determinant C, enters into the multiple summation (integration) to the inverse ! power. If it can be shown that the determinant of one type is always greater than that of another, then it follows that the contribution of the former must be smaller. Now all Class II determinants appearing through the fourth Term have been found to contain, when expanded, a net number of positive terms in addition to those contained in the expanded forms of C21 and C/, respectively. That is term for term the Class II determinants are always larger than the corre­sponding Class I determinant. Furthermore, this has been found to be true for every Class II determinant which has been studied in Terms beyond the fourth.

More particularly, if we confine our attention to those general types of Class II having only nearest neighbor intersections in their representative diagrams (inter­sections of the dotted lines) and where such intersec­tions are not adjacent nor appear at the bottom or top of the diagram, then more detailed conclusions are suggested. If we tilt a diagram on its side, erase the two solid lines and make the dotted lines solid we may symbolize the general type just referred to by IIXIIXIXI· .. IXIIIXII. (Figures land 2 contain no such types.) The corresponding representative permu­tations may be generated by applying nonadjacent nearest neighbor transpositions to the ordered number set. Now C.I when expanded contains 28 terms each containing s factors. It can be shown that the introduc­tion of a single nearest neighbor transposition introduces 28-1 positive terms (of s factors) in addition to C.I (and no negative terms). The introduction of a second (non­adjacent) nearest neighbor transposition introduces

19·2,-3 terms in addition to CI, etc. Thus, not only do such types in general appear to contribute considerably less than does the corresponding no-overlap type but as more and more transpositions (looping back) are intro­duced this difference would appear to become gradually larger. Types in which one dotted line intersects with more than one other are much more difficult to discuss on this basis. A similar trend appears to exist for them as well however. In those types having a small number of intersections compared to the total number of contacts, it is the former set of types that predominate, and the suggested trend is certain to exist.

To generalize then from this qualitative analysis of these special types, one might suggest that those types whose expanded determinants contain the identical number of net positive terms will have more or less the same average contribution. Furthermore, this net num­ber of positive terms will be directly proportional to the number of intersections appearing in the representative diagram Cor to the number of inversions contained in the corresponding permutation).16 We now state two pro­posed rules:

(1) Class II types have a considerably smaller con­tribution, type for type, than do the Class I types.

(2) The contribution of a given type of Class II varies inversely with the number of inversions contained in the representative permutation or, equivalently, the number of intersections occurring in the corresponding diagram.

Physically, this does not appear unreasonable. The greater the number of inversions the greater is the degree of looping back at different spots to achieve contact. Such looping back would seem to be less probable on the average than the simpler "side by side" multiple contact of the no-overlap type (diagrams lA and 2A for example).

It must be pointed out that all of the present argu­ments apply only up to k(k-l)/4 inversions. Beyond this the molecules are becoming untangled again until after k(k-l)/2 inversions we have come to the second no-overlap type (diagrams 1B and IF of Figs. 1 and 2).

Rule 2 leads one to distribute the k! types according to the number of intersections contained in their repre­sentative diagrams. This corresponds to the distribution of permutations by number of inversions. The gener­ating function for the distribution is

l

II (l-x i)/(I-x), (32) i=l

where the coefficient of xi gives the number of permuta­tions (containing k numbers) having j inversions. The asymptotic form of this distribution is knownl7 to be

u(k,j)=k!(n)-t exp(-u2/r), (33)

16 That the number of intersections is equivalent to the number of inversions in the representative permutation is shown, for example in A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Ltd., London), p. 32.

17 H. G. Haden, Proc. Cambridge Phil. Soc. 43, 1 (1947).

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SEC 0 N D V I R I ALe 0 E F FIe lEN T 0 F POL Y MER SOL UTI 0 N S 1009

where

u= j-[k(k-1)/4], r=k(k-1)(2k+5)/36.

Formally, we may write then for the kth Term (k very large) .

B k = T k f'" f(k,j)u(k,j)du, (34) -00

where f(k,j) must properly express that fraction of T k

which represents the contribution of a type having j inversions. According to rules (1) and (2) as well as the discussion, the function f(k,j) must sharply decrease from unity (at j=O) then gradually diminish further until it reaches a minimum at j=k(k-1)/4.lt must be symmetrical about j=k(k-1)/4 according to the pre­viously discussed symmetry in the problem. For j not zero (or k(k-1)/2), f(k,j) can be expected to be a decreasing function of k. If it is assumed that f(k,j) is unity for all j (and k) then one obtains

Bk=k!Tk•

If Class II is considered negligible, so that f(k,j) = 1 for j=O and j=k(k-l)/2 but zero elsewhere, then

Bk=2Tk,

thus establishing the limits given in (31). If, in addition, we invoke the physical requirement that the series for h(z) must converge for all z, the upper limit of (31) can be replaced by the statement that the ratio Bk/Tk may not vary more rapidly for large k than r(s'k), where O<s'<!.

IV. COMPARISON OF THEORIES FOR THE REGION OF SMALL INTERACTIONS

The expressions for A 2 as derived by the two smoothed­density models; the Gaussian density body (FK)6 and the summed Gaussian body (IK),! may be written correctly to the third term as

A 2FK=(NoX/2Mo2)(1-0.918z+0.867zC .. . ), (35)

A2IK= (NoX/2Mo2)(1-1.043z+1.135z2 - •• • ), (36)

where it is only necessary to write x= 3~z and,8= X to go from the notation used by FK and lK to that used here.18 These expressions are to be compared with Eq. (26). For vanishing z all equations reduce to Eq. (17) with H set equal to unity (the single contact approxima­tion) as they must. However, when z does not vanish, yet is sufficiently small (say, O<z<O.l), so that Terms of o (Z3) are unimportant there is considerable disagree­ment between the smoothed-density models and the present random flight model. If for example we write in Eq. (26) and (35) X = (21l"b2/3)1zn-!, then prime the z's appearing in Eq. (35) and, finally, equate (26) and (35)

18 The z's appearing in (35) and (36) are equivalently defined through (21), (10), and (8).

then we find that correct to the third Term

z(1-2.865z+9.726z2 - •• • )

=z' (1-0.918z' +O.867z'2- ... )

or on expanding z as a function of z'

z=z'(1+1.946z'+2.297z'2+ .. . ). (37)

Thus in this region the z obtained experimentally by the present theory may be measurably larger than that obtained when using Eq. (35).

On the other hand, the same parameter z appears also in intramolecular excluded volume theories. Stock­mayerl9 has compared the theories based on the two models and finds in effect that

z=2.04z', (38)

where again the primed z refers to the smoothed-density z, That is, measurements based on the intramolecular excluded volume effect (viscosity or light scattering) will give with the random flight theory9 a value for z approximately twice that obtained when the smoothed­density theory is used.20 We may conclude for conditions where higher Terms are negligible that independent measurements of a (intramolecular expansion factor) and A2 will lead to two different sets of z's depending on the theory used. At the same time there appears a tendency for both theories to be internally consistent. That is, if one theory obtains similar values for z from the two independent sources then so will the other. This internal consistency must break down in one or both of the models for very small values of z. The work of Orofino and Flory indicates good internal consistency for the smoothed-density model for values of z larger than those considered in the present discussion. They find that agreement in predicted z's is diminished if the smoothed-density theory is used for A2 and the random flight theory for (12. This clearly can also be expected for the region of small z here being considered. Stockmayerl9

has given an example at small z where the random flight model is internally consistent. Evidently a sensitive test of the theories would be provided by a detailed and specific study of A 2 and (12 at low z.

V. RANDOM FLIGHT MODEL MODIFIED TO INCLUDE INTRAMOLECULAR EXCLUDED VOLUME

When it is necessary to consider the higher terms of the general development (16) one should properly choose a model which takes into account intramolecular ex­cluded volume effects. While the object of this study has been to investigate the random flight model in the general sense of (17) it is of interest to present, without evaluation, the nature of the development through the third Term when the random flight model is modified to include intramolecular excluded volume. Following the treatment of the intramolecular volume effect by

19 W. H. Stock mayer, J. Polymer Sci. 15, 595 (1955). 20 P. J. Flory, J. Chern. Phys. 17,303 (1949).

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1010 ANDREAS C. ALBRECHT

Fixman,9 the distribution function (18) is modified to give

F1'(1)r=Fl (1)I II<U (l+Xii) / , 1

where the Xi; relate now to potentials of average force between two segments of the same molecule (we shall use Roman letters when segments of the same molecule are considered to interact).

The development of (39) takes the form of an expan­sion in terms of increasing numbers of simultaneous intramolecular contacts. Now, correct to the single contact approximation, A2 is independent of model. Thus the effect of intramolecular excluded volume can­not be felt until at least two intermolecular contacts have been established. The earliest modification of the Az development is therefore not to be found before the triple contact Term (2 inter- and 1 intramolecular con­tact). Straightforward application of (39) after the method of Fixman modifies the triple contact Term of (26) from 9.726z2 to

9. 726z2+ 2n-3 (27rlr j3)3Z2

L [Pz(0I'.,Oii).X-P1(0I',).XPl(Oij)]. (40) K.,).,Jl" v,i,i

With arguments exactly analogous to those used when discussing the relative magnitudes of the Class II types, it is not difficult to show that the argument of the multiple sum is always positive. Hence intramolecular excluded volume increases the magnitude of the third Term. This is in qualitative agreement with what one would obtain by writing z/a.3 in place of z. This latter substitution, using9 (i=1+2z-···, would give 5.731 for the magnitude of the correction, giving finally

A 2= (NoX/(2Mo2»(1-2.865z+15.457zL . .. ) (41)

but this probably represents an overcorrection. The integrations required to evaluate the correction exactly appear to be very lengthy and have not been carried out.

VI. REMARKS ON THE ASYMPTOTIC BERA VIOR OF A2

Unfortunately, the present development of the ran­dom flight model is not sufficient to permit a rigorous theoretical discussion of the behavior of A2 at very high molecular weights. The upper and lower limits found in (31) for the coefficient of Zk in h(z) leave much too broad a gap. Complete evaluation of the Class II types would be of great value, but without proper treatment of the intramolecular interactions even such progress would be insufficient.

It is nevertheless of considerable interest to explore the consequences of the strong probability (Appendix D) that h(z) asymptotically becomes C/z, where C is a

positive constant. To introduce as best we can the intra­molecular effect, we replace z by z/a3 as in Sec. V, and then obtain (for large z)

A2=NoXCa.3j2Mo2Z. (42)

If we introduce the definition of z from (21) and the mean square displacement length (V)=nb2c?, this rela­tion may be written

AzMz= (27r/3)!CNo(V)!/2, (43)

which states that for high molecular weights the molar second virial coefficient A 2M2 is simply determined by an average volume of the coiled macromolecule. Since the intrinsic viscosity is given by the well-known Flory­Fox relation

(44)

we may form the dimensionless ratio

A zM/[l1] = (27r/3)~CNo/2<I>=const. (45)

Now exactly this relation, with the dimensionless con­stant having a value of about 1.4±0.2, has been ob­servedzl- z4 to represent experimental results for various linear high molecular weight polymers in good solvents (Le., high z). Such a relation is of course evident for hard spheres of volume v, for which AzM2=4v and [11]M=5v/2, giving A zMj[l1] = 1.6; and it also results at high molecular weights from Flory's first dilute­solution theory25 (swollen spheres of constant density throughout). If in the present theory we take the value C=O.276 from (D-8), in which all Class II types are neglected and only the asymptotic coefficients T k * are used and then insert the preferred26 experimental value <I>=2.5XlO23 , we obtain A 2M/[11]=1.0. This is grati­fyingly close to the experimental figure, especially in view of the crude way in which intramolecular inter­actions are taken into account.

The Gaussian smoothed-density models lead to less satisfactory behavior at high molecular weights. Thus, Isihara and Koyamal found from their function that A2 varies approximately as the inverse 0.23 power of M at high M, in apparently good agreement with the experi­mental results. However, since they took no account of intramolecular effects, this agreement must appear for­tuitous. In fact, if again one writes z/a.3 in place of z, the agreement is destroyed, as shown by Krigbaum and Floryz7 for their model.

21 E. F. Casassa, thesis, Massachusetts Institute of Technology, 1953.

22 A. R. Shultz, J. Am. Chem. Soc. 76,3422 (1954). 23 S. N. Chinai and R. J. Samuels, J. Polymer Sci. 19, 463 (1956). 24 Atkins, Muus, Smith, and Pieski, J. Am. Chem. Soc. (to be

published), results for linear polyethylenes. 25 See reference 1. It is interesting to note that for small z this

theory giveslO (in our notation, and if we choose R2=nb2/6)h(z) = 1-1. 72z+ ... , which is in better agreement with (26) than the Gaussian smoothed density models.

26 Newman, Krigbaum, Langier, and Flory, J. Polymer Sci. 14, 451 (1954).

21 W. R. Krigbaum and P. J. Flory,]. Am. Chem. Soc. 75, 1775 (1953).

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SEC 0 N D V I R I ALe 0 E F FIe lEN T 0 F POL Y MER SOL UTI 0 N S 1011

VII. CLOSED EXPRESSIONS FOR A.

It would be very desirable to have a simple expression in closed form to represent A2 at all molecular weights in any solvent. This would permit deductions from experi­mental data of unambiguous thermodynamic interaction parameters, in a way not possible at present when we have no equation which adequately fits the observed dependence on molecular weight over the whole range. In this connection Orofino and Flory6 have found that the FK theory can be well represented by an expression of the type (in our notation)

(46)

where C is an appropriate constant. If we choose C=5.73 and expand (46) for small z we obtain

h(z) = 1-2.865z+16.630z2- "', (47)

which compares favorably with (41). A second simple closed form, first proposed by

Stockmayer28 reads

h(z) = 1/(1+2.865za-a),

which for small z yields, on expanding,

(48)

h(z)=1-2.865z+13.938z2 - ••• , (49)

also in good agreement with (41). Furthermore for high molecular weights (48) leads to A 2M/['I1J'""1.3 in excellent accord with experiment. Equation (46) varies more slowly with M and therefore does not predict A 2M/['I1J = (constant) at high molecular weights. Orofino and Flory have applied (46) (with C=2.30) to a wide variety of data with considerable success in bringing out the self-consistency of the smoothed den­sity model as has already been mentioned. Equation (48) has been tested successfully in several cases.28

Although (46) and (48) do not yield series which converge for high z, they may still offer satisfactory approximations to the exact function over a wide range of the variable.

VIII. CONCLUDING REMARKS

The present work has taken the theory of A 2, based on the detailed analysis of the interaction of random flight chains, to what appears to be a practicable limit using ordinary nonmachine methods. The possibility, how­ever, that a good approximation to the function f(k,j) [Eq. (34)J may be deduced, cannot be discounted. And should the arguments which lead to Eq. (34) prove reasonable then an asymptotic form based on the pure random flight model may yet be achieved. Nevertheless, as has been argued, an asymptotic form which does not take into account the effects of intramolecular excluded volume cannot be a realistic one. The formal inclusion of this effect into the random flight model represents a problem of an even higher order of difficulty.

28 J. P. Bevak, thesis, Massachusetts Institute of Technology, 1955.

We may also remark that the complexity of the problem is still further increased when the interaction of two chains of different lengths is considered. The double­contact Term is easily evaluated in this case, but is already an unappetizing if elementary function of the two lengths. For this reason, we have not considered the effect of molecular-weight heterogeneity on A 2, which has been treated for the smoothed-density model by Flory and Krigbaum. Db

For small values of the parameter z the present theory can be tested given suitable data. If with some minor approximations a few of the next higher Terms can be evaluated then the domain of applicability would be extended. However, for large z it would seem that one should turn rather, to a refinement of the smoothed density models by including explicitly at least a few internal coordinates (fluctuations of the smoothed densities) of the bimolecular system, as has already been proposed,s

APPENDIX A

Fixman,9,10 by generalizing to three dimensions a derivation of Wang and Uhlenbeck,13 proves the following:

Given s vectors V 1, V 2, • , " V. each of which is a linear combination of the n Gaussianly distributed segment vectors r1, r2, "', r n where for example

(A-l)

then the simultaneous probability distribution of V 1, V 2,

"', V. is

P.(V1,· •• ,V.) = (3j27rb2)MC.-!

Xexp{ - (3/2b2C.) L L CiiVi'Vj } , (A-2) i i

where Cii is the cofactor of the element C if of the S2

matrix [C i )] where n

C;j= L iftitiftjt, t=1

and C. is the determinant of [C;j]. It follows in the present problem that, since a loop represents a vanishing linear combination of the 2n segment vectors r1) r2, .. " r2n with coefficients zero or unity,

APPENDIX B

The integrals (24) and (2Sa) are special cases of the general integral, T k, described in Appendix C. For k= 2 we find

2n! f1J1 (1- L)(l-S) (L+S)-JdLdS o 0

= 2(1.4327)n i (B-l)

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1012 ANDREAS C. ALBRECHT

and for k=3

2n3 (j.l1L IS (I-L)(l-S)(L+S-XI-Yl)-! Jo 0 0 0

X (Xl+Yl)-!dLdSdxldYI = 2 (1.8997)n3. (B-2)

The integral of (2Sb) proves to be a much more difficult one. Only after extremely laborious calculations which cannot be outlined here was it possible to obtain the evaluation which is correct to the precision of the graphical integration performed. It was found that

1 1 LiS 4n3 r I I (I-L)(I-S)[(L+S-xI-YI)

J 0 0 0 0

X (Xl +YI) + XIYI]-JdLdSdxldYI = 4 (1.482)n3. (B-3)

APPENDIX C

The integral, T k may be written

where ill

T k= f (l-L)(1-S)Jk(L,S)dLdS, o 0

~x,=L ~Yi=S

(C-1)

(C-2)

L, S, Xi, and Yi are reduced variables as defined in text15 and in Figs. 1 and 2.

The integral J k has been carried out through k= 6. It is possible to write

where Land S of the k-1 multiple contact become the X and Y of the kth multiple contact. A very useful change of variable proves to be

x=r2(1-u)/u, y=r2,

which gives however an answer not symmetric in Land S. This is made symmetric by adding an equivalent term and dividing by two. The answer, generalized, is found to be

J k(L,S) = 2k-2(k_1)7r(k-2)/2

X (LS) (k-2)/2(L+S)-(k+l)/2/I' (k/2). (C-4)

This is now substituted into (C-1) and the final integration over Land S is carried out. Expanding (I-L)(l-S), one finds that

T k= const.(k)[Gk_2, k+l-G k- 2, k-l+Gk, k+l], (C-S)

where

(C-6)

The integration (C-6) is readily achieved through the substitutions L=r2 tan20, S=r2, giving

T k= 2k+27r(k-2)/2[ (k2+3)I k-l- (k-1)2-(k-l)/2]/

(k+ 1) (k+3)r (k/2), (C-7) where

(C-8)

The recursion relationship for In reads

n1n= (n-1)In_2-2-n/2• (C-9)

When (C-9) is expanded one obtains for odd n,

In= (n-1)(n-3) .. '2[1_2-~_'" n(n-2)·· ·3·1

(n-2)(n-4)···3 J _ 2- n /2 •

(n-1)(n-3)·· ·2 (C-lO)

The negative terms in (C-lO) represent the first n terms oftheexpansionof xd(sin-1x)/dx evaluated at x= (1/2)1. This latter expansion is just unity. Hence, the entire bracketed term of (C-lO) represents the remainder of this expansion after the first n terms. It is possible then to write

_ 1 (1)(n+2)/2[ n+2(1) In-- - 1+- -

n+1 2 n+3 2

+ (n+4)(n+2)(~)2+ .. 'J, (C-ll) (n+S) (n+3) 2

which gives finally after approximating the term in the brackets (of C-ll) by (1-! )-( n+2/ n+3)

I n"-'2-(n+2) (n+1)/2(n+3)/ (n+ 1). (C-12)

This may be similarly obtained for n even. This ap­proximation is found to be good to about 1% already for Io. Putting (C-12) in (C-7) one obtains

(k+ 1) (k+3)r (k/2)

X [(k2+3)2-1/(k+2)/k-k+ 1] (C-13)

and for k sufficiently large

T k *""2!(1-ln2) (27r)k/2/7rr (k/2+ 2). (C-14)

APPENDIX D

We here attempt a discussion of the behavior of the series

'" h(z)=I+ L Bk(-Z)k-l (D-1) k=2

for large z. From the asymptotic formula (C-14) for Tk

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SEC 0 N D V I R I ALe 0 E F FIe lEN T 0 F POL Y MER SOL UTI 0 N S 1013

it is easily seen that the upper limit of (31) for B k would give a divergent series. In view of the discussion of Sec. III, the lower limit is probably closer to the true situation.

If the asymptotic value Tk* from (C-14) is adopted for all Tk and the lower limit of (31) is employed, we have the function

00

h*(z) = L 2Tk*(-z)k-l=47r-!(1-ln2)S(x), k~l

(D-2) 00

Sex) = L (-x)k-1jr(kj2+2); x= (211')lz. k~l

The series Sex) is seen to be absolutely convergent, and thus may be split into two parts:

S(X)=Sl-S2,

00

SI= L rmjr(m+!), m~ (D-3)

00

S2= L x2m+ljr(m+3)=x-3[exp(x2)-1-x2]. m=O

To evaluate SI, we write

00

x3S1 = f(x) = L x2m+3jr(m+!) (D-4) m=O

and note that f(x) satisfies the differential equation

f'(x)- 2xf(x) = 411'-!r,

which integrates to

f(x) = erfx expr- 211'-!x. (D-S)

We therefore obtain

S(x)=x-L 211'-!x-2+x-3(1+erfcx expx2) (D-6)

and by using the asymptotic formula for the comple-

mentary error function we find

S(x) = X-1+0(X-2) , (D-7)

which corresponds to the asymptotic result

h*(z)= 2!(1-ln2)jll'z= 0.276jz. (D-8)

Now we may not identify h*(z) with the true h(z), even if we assume that B k= 2T k, because (C-14) is only asymptotically correct. However from the form of (C-7) or (C-13) we may infer that the correct Tk can be represented to a satisfactory numerical accuracy by a finite polynomial of the form

m

T k = (211')k/2 L A nkjr(kj2+nj2), (D-9) n~

in which case if we do assume that Bk=2Tk it can be shown that h(z) must behave asymptotically as (constl+const2jZ). Yet the contribution of Class II types has until now been entirely suppressed. The arguments in Sec. III, 4b show that type for type the Class II types converge more rapidly than does T k

(type I) although the number of the former increases rapidly with k. If as a simplifying assumption we say that the increase in number of Class II types does not overwhelm their individually more rapid convergence and that therefore their entire contribution converges at least as rapidly as does Tk then we may conclude that the true h(z) goes asymptotically as (const3+const4jz). Finally we appeal to experiment which indicates that h(z) goes asymptotically to zero. The difference then between the true h(z) and h*(z) can give no constant or divergent remainder. The consequences of this asymp­totic behavior are discussed in Sec. VI.

ACKNOWLEDGMENT

The author wishes to express his gratitude to Pro­fessor W. H. Stockmayer for the numerous suggestions and stimulating discussions which served both to inspire and enrich the present investigation.

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