Conformational analysis of polyethylene and isotactic polypropylene

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CONFORMATIONAL ANALYSIS

OF POLYETHYLENE AND

ISOTACTIC POLYPROPYLENE

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGE-SCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS

IR. H. J. DE WIJS

HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 8 DECEMBER 1965 DES NAMIDDAGS TE 2 UUR

DOOR

ABRAM OPSCHOOR

1965

UNIVERSITAIRE PERS ROTTERDAM

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. W. PRINS

Aan de nagedachtenis van mijn vader. Aan mijn moeder.

CONTENTS

INTRODUCTION 1 I. CHAIN S T A T I S T I C S 3

1.1 Introduction 3 1.2 Chains with independent rotations 4

1.3 Chains in which the rotations are influenced by the nearest

neighbours 4 1.3.1 The transformation matrix 5

1.3.2 Calculation of the partition function of the polymer

chain 7 1.3.3 Averaging of the functions of rotation angles 9

1.3.4 Averaging of the matrix products 10 1.4 Chains in which higher order interactions occur 12

1.5 Isotactic chains 15 II. C A L C U L A T I O N OF THE E N D - T O - E N D D I S T A N C E

FOR P O L Y E T H Y L E N E CHAINS 1 7

2.1 Introduction 17 2.2 Functions relating geometric variables to energies 19

2.2.1 Interactions between non-bonded atoms 19

2.2.2 Distortion of bond angles 20 2.2.3 Changes in bond lengths 21 2.2.4 Rotations around the chain bonds 22

2.3 Possible chain conformations and their energies 22

2.4 Calculations of the end-to-end distance 27

2.5 Form of the chain 29 III. I S O T A C T I C P O L Y P R O P Y L E N E 3 1

3.1 Conformations of the chain 31 3.2 Calculations of the end-to-end distance 33

I V . E X P E R I M E N T A L VALUES OF THE E N D - T O - E N D D I S T A N C E 3 7 4.1 Evaluation of the end-to-end distance from viscosity

measurements 37 4.2 Evalutation of the end-to-end distance from Ught scattering

measurements 42 4.3 The temperature coefficient of the end-to-end distance 44

Aan de nagedachtenis van mijn vader. Aan mijn moeder.

V. RUBBER ELASTICITY AND CHAIN CONFORMATIONS 4 5 5.1 Introduction 45 5.2 Apparatus 48 5.3 Materials 50 5.4 The measurements 52 5.5 Results 54 5.6 Conclusions 59 SUMMARY 61 APPENDIX 1. Specification of the energy values in section 2.3 63

APPENDIX 2. List of important symbols 66

The author is much indebted to the board of the Algemene Kunstzijde Unie N.V. and to the management of the Central Research Institute of the Algemene Kunstzijde Unie N.V. for offering the opportunity to carry out the in-vestigation described in this thesis.

Thanks are due to all people, who have contri-buted in any way to this thesis, in particular to the co-workers at the Laboratory of Physical Chemistry, Technische Hogeschool, Delft.

INTRODUCTION

In the past decades research on macromolecular materials has developed rapidly. In many cases this development has taken place under the direct influence of technology in which an extensive use is made of macromole-cular materials such as plastics, resins and rubbers. The experimental and theoretical investigations are mainly aimed at finding connections between the physical properties of these materials and their structure in the crystal-line and amorphous state. The primary object is to obtain the best possible insight into the molecular structure.

In this investigation special emphasis is laid on the determination of the unperturbed dimensions of the molecules of polyethylene and isotactic polypropylene such as occur in certain solvents and as are assumed to occur in the molten state, and on the determination of the form of these chains. It should be noted that, owing to the possibilities of rotation around the chain bonds, a polymer chain can assume an enormous number of conformations. Thus, one can only speak of the average dimensions of the chains, as characterized e.g. by their reduced mean square end-to-end distance, <.rh>/nl^. This dimensionless quantity is the mean square of the distance between the ends of the chain, divided by the number of bonds, n, and the square of the bond length, /.

Many attempts have been made to calculate the end-to-end distance of polyethylene with the aid of conformational statistics. No reliable results have been obtained, however, since insufficient information has been avail-able about interactions between neighbouring bonds. In several older calculations these interactions were even completely neglected. For iso-tactic polypropylene some calculations methods have also been described in the literature.

Beside these theoretical calculations experimental techniques have been developed by means of which the end-to-end distance can be determined. The analysis of the experimental data, however, is rather involved.

Several authors have stated that it is also possible to obtain conformational data via the theory of rubber elasticity. For that purpose, force-temperature measurements at constant length must be made on strips of amorphous, crosslinked polymer.

In this investigation, use has been made of both theoretical calculations and force-temperature measurements.

Chapter I gives a further elaboration of the chain statistics involved. Special attention is paid to the matrix calculation-methods of Lifson' and Hoeve". These methods have been further extended, making it possible to take into account higher order interactions and other effects. The necessity for this extension appears in Chapter II, in which a completely a priori calculation of the end-to-end distance of a polyethylene chain is made. In Chapter III a study is made of the conformations of the isotactic poly-propylene chain. In Chapter IV a comparison of the values of the end-to-end distances, calculated in Chapters II and III for polyethylene and isotactic polypropylene, is made, with the values available in the litera-ture. Since the values in the literature are rather scattered, an examination of the reUability of the various experimental techniques is presented. In Chapter V the rubber elasticity theory and its application for obtaining conformational data are explained. Chapter V also deals with the experi-mental execution of the force-temperature measurements and with their interpretation. Measurements have been made on both polyethylene and isotactic polypropylene. Finally a resume of calculations, experiments and conclusions is given in the Summary.

Chapter I CHAIN STATISTICS

1.1 INTRODUCTION

The dimensions of chain molecules in solution are strongly influenced by interactions between the chain elements. These interactions are usually distinguished as short-range and long-range. By short-range interactions is meant interactions among atoms or atom groups, which are separated only by a small number of bonds. They determine the length of the so-called effective monomer unit b, which is defined by the relation:

<rl>=nb^ (1.1) The long-range interactions, on the other hand, are caused by groups

which, in the fully stretched state, are far apart. In the chain statistics the latter interactions are not taken into account. This may be justified as follows:

The influence of the finite volume of the chain can be accounted for by excluding all conformations in which two or more segments occupy the same volume element. The excluded conformations will often have a high degree of coiling. Consequently, there will be an increase in the average dimensions of the chain due to the volume effect. However, in addition to interactions between chain segments, interactions between chain segments and molecules of the solvent will occur. In poor solvents the polymer will repel the solvent and will contract. Under these circumstances the con-traction of a macromolecule can, at a certain temperature, just compen-sate the expansion due to the volume-effect. The macromolecule then has its ideal dimensions and is called unperturbed. The temperature at which the molecule has its unperturbed dimensions, is called the ©-temperature. This point is analogous to the well known Boyle-temperatures of real gases, where a gas behaves like an ideal gas.

Flory» has pointed out that in the amorphous phase the chains are similarly unperturbed. In this case, the chain segments are siu-rounded by other similar segments. The chain cannot distinguish between the influence of the backcoiling part of its own chain and the influence of other chains and consequently, long-range interactions cannot have any influence on the chain dimensions. However, this does not mean that the length of the effective monomer unit, b, in the amorphous phase has to be equal to the

length in ©-solvents.

Kuhn^ has calculated that the average end-to-end distance for unper-turbed chains of sufficient length is proportional to the square root of the number of bonds. Moreover, the distribution of the end-to-end distances appears to be Gaussian. Only for short chains and for macromolecules in which strong interactions occur among the segments, as in the case of various helix-forming polypeptides do these relations not apply.

The statistical investigation into the end-to-end distance of the unperturbed polymer chains must be gradually developed. On the basis of the simplest chain model, on which no restriction whatever is imposed as to the direc-tion of the bonds, attempts must be made to obtain a model in which fixed valence angles and interactions among substituents such as occur in a real chain molecule have been taken into account. In the following sections some models will be discussed.

1 . 2 CHAINS WITH INDEPENDENT ROTATIONS

As early as in 1932, Eyring' indicated a method for the calculation of the mean square of the end-to-end distance for a simple model. Since then his model has been improved and extended by various authors. Benoif, Sadron' and Taylor" have calculated the end-to-end distance as a function of the bond length / and the bond angle 6, assuming a limited free rotation around the chain bonds which is symmetrical with regard to the trans position. Thus for the reduced mean square of the unperturbed end-to-end distance <_rH'>/nP, they derive:

< r o > 1 - COS0 1 + <cos<j!)> ,. nl^ 1 + COSÖ l - < c o s ^ >

In this formula, <cos 9P> represents the average cosine of the angle of rotation <p. When free rotation around the chain bonds is possible, the average cosine will be equal to zero. Then Equation 1.2 reduces to Eyring's original formula.

1.3 CHAINS IN WHICH THE ROTATIONS ARE INFLUENCED BY THE NEAREST NEIGHBOURS

Almost simultaneously Lifson', Volkenstein^ and Nagai» developed cal-culation methods based on the use of matrices. These methods are pre-emmently suitable for including effects due to interactions among neigh-bouring bonds. With one matrix, namely, the group of all states of a cer-tain bond or of a pan of bonds can be represented. Although for various

bonds the contributions of the separate terms in the distribution functions to be treated are different, it is one of the characteristics of a chain mole-cule that each bond or each pair of bonds has the same group of possibilit-ies at its disposal. Since in averaging, the total group of possibiUtpossibilit-ies of each bond is important, the matrix representing this group is suitable for the calculation. These matrix methods have been introduced by Kramers and Wannieri» in order to solve the Ising problem. In connection with the extension of the calculation in Section 1.4 the method of Lifson^ and Hoeve" will be further discussed.

1.3.1 The transformation matrix

To each bond vector is assigned a rectangular coordinate system. This is done in the manner indicated by Hoeve".

The z„-axis coincides with the bond vector L. The y„-axis lies in the plane through Iv and lv_i in such a way that the positive direction of y« forms a right angle with L^i. The Xv-axis completes a right-handed coordinate system. Thus in its own coordinate system each vector is represented by the vector {0,0,1}. The matrix T„ which transforms the (v + l)-th system into the v-th is essential. It can be obtained with the aid of two successive

Figure 1.1 Rectangular coordinate systems assigned to the bonds /„ and l^^^. The bonds / „ J and /„ lie in the plane of the paper.

elementary transformations. From Figure 1.1 it follows that first the (v -\- l)-th system must be rotated in a negative sense round the Xi, + i-axis through an angle of 180 — 6. Then a rotation through an angle of

y;

Figure 1.2 Positions of the axes after the first transformation.

The product of the matrices Tg?,, and T$ corresponding to the elementary transformations is equal to the matrix

— cos<p„ COSÖ sin<p„ sinö sin^„"

sincp^ C0SÖ cos^„ sin0 cos<iff„

0 sin0 — COSÖ

It is possible, by repeated application of transformations, to express the coordinates of 1« in terms of those of the first system.

l = Ti T„-i {0,0,1}

When all vectors have thus been defined with respect to the first system, the end-to-end distance is given by the following vectorsum

To = (E3 -1- T i + T1T2 + + T , T„-i) {0,0,1}

The square of the end-to-end distance is obtained by multiplying to by its transposed r*o. Carrying out this operation, it should be remembered that Tv is an orthogonal matrix, so that T*„T^ = E3. Moreover, the only element of the resulting matrix that is important is the (3-3) element. Therefore the transposed matrix T*» may be replaced by T»

rg//^=[nE3+2{(Ti+ . . . + T„_ 0 + (TiT, + . . . +T„_,T„_,) + +(TiT2 T„_i)}]3.?

<r?>/Z^ = [nE3 + 2{(n-l)<T„> - f - ( n - 2 ) < T X + i > + •••

+ <Ti T„_i>}]3.3 (1.3) For long chains the above formula can still be considerably simplified,

since the elements of the transformation matrix are all smaller than one. Therefore, as the number of factors in a product of the matrices increases, the numerical value of the 3,3 term of the product decreases rapidly. Consequently, the series in brackets may without any objection be re-placed by an infinite series. Moreover, the coefficients of the terms of the series may be replaced by n. Equation (1.3) can then be written as follows:

<rl>lnl'- = [E3 -t- 2{<T„> + < T X + i > + }]3.3 (1-4) The mean value <Tt,> of the transformation matrix T„ is found by averaging all the elements of this matrix. This average is independent of V when the chain consists of one type of bonds.

The averaging of the products of the matrices requires knowledge of the partition function of the polymer chain and of the way in which functions of rotation angles must be averaged.

1.3.2 Calculation of the partition function of the polymer chain

As a first approximation the energy, e'^^h, ;• , of a bond is provisionally made dependent only on its own condition of rotational state and that of its nearest neighboiu-s. The superscript v of £ indicates the v-th bond, the subscripts / and ƒ specify respectively the rotational angles of the v-th and (v-f l)-th bond. The probability of the bond-pak being in this state, is given by

wj:]=exp(-4:]ikT)

The probability that a certain conformation of the chain occurs is given by the infinite product:

' . J J i t y,2 ''z.i

To this product one extra term has been added, which means that it is assumed that interaction also occurs between the last and the first bond. As a result, the system becomes closed. When n is large, the influence of the extra term is negUgibly small.

con-formations. This summation, however, is only practicable when tp assumes only a certain number of discrete values: q>i., <pi.. . <pm- The magnitude of m is determined by the form of the rotational potential and the accuracy desired.

ö = E

w. (»)

As iv(''>j,; is a function of two variables, it is possible to combine all its values in a matrix w., w, w, w, l _ " ' m l ^mn

This matrix is independent of v, because it contains all the values that Wi,j can assume.

On multiplication of the matrix Wi,2 by the matrix W2,3 we find for the element (/, k)

J = l

Each element of the matrix product is obtained by summing the products of the right-hand part over each q}, except for the fkst and the last. More-over, use can be made of the relation

(Wi,2W2,3W3,4),-,, = (W^),-,, since w is independent of v. Thus:

i . . . I wi^>... ww=(w,:,)

J = l z = l

and therefore one has for the partition function

m

0 = I ( W ) u = trace(w")

i = l

Through a similarity transformation the matrix w can be transformed into a diagonal matrix A, which has the characteristic values of w as elements.

The matrix x contains the characteristic vectors of w. The vectors occur in the same sequence as the corresponding characteristic values in the diago-nal matrix. In this transformation the trace of the matrix remains un-changed. Since n is very large, the smaller characteristic values in the partition function can be neglected in comparison with the largest, X, and a simple equation for the partition function is found:

Ö = A" (1.5)

1.3.3 Averaging of the functions of rotation angles

Generally, the average of a function of a rotation angle, f95 <^') is given by

f(<p"")=^t •••t<j---<r''f(<?'J''Vïï• • • wir/ (1.6)

't.--This summation becomes practicable by defining a diagonal matrix F which contains all the values that f(^<''0 can assume and consequently is independent of v.

n(<Pi"") 0 -] F =

L 0 f(<pr)j

As has been done for the partition function, it can be deduced that Equation (1.6) changes into

<f((p)> =itrace(w''-'Fw"-"+i)

e

The trace does not change through cyclic permutation of the matrices <{((p)> = - t r a c e (Fw")

Q

Subsequently, similarity transformations are applied to w and F. This successively yields:

< {((o) > =~ trace (x" * F x A")

e

<f(,?))>=iXZ(x"'rx),,(A"),,

Since {A.^)j,i — 0 if j y^ i, this becomes <f(<?>)>=iX(x-'Fx),,(A'')M

6 '

which may be written as

<f(<p)>=iXxr'Fx,(A%

Ö '

where Xj is a column vector of x.

In this equation again only the largest characteristic value is of impor-tance, so that finally

<f(cp)> = 'LVn^" = 'i'n (i.7a)

Ö

Here |* and i are the left-handed and the right-handed characteristic vectors belonging to the largest characteristic value X of the matrix w. The characteristic vectors must have been normahzed in such a way that

é * l = L

The averaging of the products of two or more functions of successive angles takes place similarly. The following formulae are obtained

<{{cp^"^)g(<p^''^^^)> =r^'i*FwGi (1.7b) <f((?)(">)g((p<''+i>)h((?)<"+'>)> = A - ^ i ' F w G w H ^ (1.7c)

and so on.

G and H are diagonal matrices, defined in the same way as F. 1.3.4 Averaging of the matrix products

The averages of the matrix products in Equation (1.4) can be calculated when the probabilities are known with which certain successions of rota-tion angles occur.

If we designate by ai(v) the probability that the bond v has the rotation angle (pt and by ai,j(v,v-\-1) the probability that the bonds v and v-f-1 have the rotation angles 9?; and cpj, then we have

<T„>=^a,.(i;)Ti

i

<T„T„+i> =EZfli,;(t'.t; + l)T(T,. (1.8)

These probabiUties can be calculated with the aid of the formulae derived in the preceding section. To that end functions f(<p^^^), g(9''^"'"*'). etc. are defined, which are always zero, except when g? has a certain value.

f((p('") = 1 when cp^"^ = cp^f g(<p<''^i^) = lwhen<?)<"+'>=<?>j''+^>

The matrices F, G, H, etc. which contain all the values of the functions t(<p^^^), g(<35<''+*0 etc. will therefore consist almost entirely of zeros. Only in the positions (if) for F, (jj) for H, we will have unity. Then ai(v) =

<f(^)> and ay(v,v+l) = <f(9)<''>)g(9?<^+i>)>. Apphcation of the For-mulae (1.7) yields:

a,/t),i;+1) = A " ^ ^ f* Wjj. ^j etc.

After substitution of these values in the Equations (1.8) it is foimd that < T „ > = i ^ ; T , ^ i

I

<T„T„+i> = E E r ' ^ ; T i W y T , | , etc.

• J

These summations can be easily carried out if Tj, T,- etc. are scalars. The first summation would then represent the multiplication of the row vector J* by the diagonal matrix T and by the column vector | . However, now that Ti and T,- themselves are already third order matrices, the scalar quantities must first be transformed into third order matrices so as to be equivalent. This can be done without any objection by multiplying them by the third order unit matrix Es. The summation can now be carried out if the matrices D, X*, X and W are defined:

D = Ti O.E, X = • ê i E 3 W L 0 . E 3 . . . . T „ _ L^„E3J X''=[^i*E3 C E S ] W11E3 Wi„E3 - W „ i E 3 W„mE3_

The sub-matrices may be treated as common matrix elements. The averages are found to be

<T„> =X*DX

<T„T„+i> = A - ' X * D W D X

Substitution of these results in Equation (1.3) yields:

<rl>lnl' = [E3 + 2 X*D{E3„ + S -h S^ -h S^ + }X]3.3 in which S = A"' W D

After summation of the series, this formula changes into

<r'o>lnl' = [E3 + 2X* D(E3„ - S)"' X]3.3 (1.9) Thus a formal answer has been given to the question as to the extent of

the end-to-end distance of a polymer chain, taking into account the nearest neighbour interactions. Simplification is in general not possible and more-over not necessary, since the formula is quite suitable for numerical cal-culations with a computer.

1.4 CHAINS IN WHICH HIGHER ORDER INTERACTIONS OCCUR

Investigation of the polyethylene chain shows that higher order inter-actions cannot be neglected in the calculation of the end-to-end distance (see Chapter II). For the purpose of including these higher interactions in this investigation, the above treatment has been extended.

If, in addition to the influence of the first neighbour, the influence of the second is also taken into account, the energy of the v-th bond, e^^h,i,k, becomes a function of three successive rotations. The partition function is then found to be

e = I . . . I w'iVwi^» <:''<r'-^it

i = l z = l

To this sum two terms have been added to make the system closed. Al-though w^") is a function of three variables, it is possible to put all the values of w'^' m a two-dimensional matrix. The composition of this matrix must be such that its rows correspond to possible values of qj^"'', 93<"+*>, and its columns correspond to values of q}^^+^\ q)^^+^K Thus the matrix will be of m^ order. As a result of this definition, the matrix will contain a great many zeros, since all places for which it is required that 9j<^+i> simultaneously occupies two different values, have of course a probability of zero.

w =

Will o O ^211 o O _{w, m i l} 0

o

Wiim O 0 W2i„ 0 o W„i„ . .

o Wi2i . . . o o W221 . . . o o W„21 . . o o Wi2„ . . . o o W22„ . . . 0 o W„2„ . . o

o

o

o

This matrix can be considerably simplified by introducing the sub-matrices w'l, 1 w'l, 2, . . . Y/'mm of ordcr m. These sub-matrices are defined as

follows: Wo = •0 0 Ow.jiO . . . 0" • Ow,;„0 0

With these substitutions, the matrix w assumes a form which is identical to the form of the matrix w defined in Section 1.3.2:

w =

w, w„

-Wlm W„„_

Since the sub-matrices may be treated as common matrix elements, the derivation given in Section 1.3.2 can be completely followed. Thus, the partition function is now also found to be

Q = r

in which X is the largest characteristic value of w.

In this case the average of a function of a rotation angle can be represent-ed in a form shnilar to Equation (1.6). If the diagonal matrix F is definrepresent-ed in the correct manner, this summation can be made exactly like the sum-mation in Section 1.3.3. As F must be combined with the matrix w, it will also have to be of order m^. Smce only m different values are possible for

matrix is represented with the aid of sub-matrices, the analogy to the previous definition appears clearly

-fXvi"')

F =

LO.E„ f(9>nJ

where i'((p^^'>d = %<^*i).E^

With the aid of this definition it can easily be seen that the Equations (1.7) retain their validity.

In averaging the matrix products, use is made of the Equations (1.7) and the definitions in Section 1.3.4. The matrices F, G, H, etc., however, now contain not only one element equal to 1, but one sub-matrix, or m ele-ments. This leads to the following probabilities:

x = 0

m - 1

a,/t),t;+l)= Y ^'^^i'+x'^i+xj+xij^-x etc.

which must be substituted in the Equations (1.8)

m - l < T „ > = 2 J Zy ii+xT^i + x^i + x i x = 0 m - l < T „ T „ + i > = 2 ^ 2 J Z J ''• ii + x'-i + x^i + xJ + x'^J + x^j + x i J x = 0

This summation is done in the same way as in Section 1.3.4. The sum-mation over X presents no difficulties, since the transforsum-mation matrix is not a function ot x (Ti^x = TO- In the diagonal matrix D, each transfor-mation matrix will occur m times owing to this extra sumtransfor-mation

| - T ( 1 )

TV

D =

r d )

The other matrices defined remain imcbanged. The result obtained ulti-mately is almost equal to the Formula (1.9) obtained for exclusively nearest neighbour interactions. Only the unit matrix Esm is replaced by the unit matrix Esm^

< r ^ > K = [E3 + 2X'D(E3„2-S)-^X]3,3 (1.10) Extension of this formula to take mto account interactions of a bond with

its n-th neighbour, can be simply made. In this case the matrices W and F will have to be of the order m». If submatrices of the order m"-» are introduced, the whole derivation takes place shnilarly and Formula (1.9), in which Esm has now been replaced by Esm", is found again. Numerical application of this formula for higher n becomes very difficult, however. Should one wish to take into account the interaction of the third neigh-bour in the case of polyethylene, in which m can be supposed to be equal to 3, a.o. a matrix of order 81 will have to be inverted. This requires the calculation of 6561 determinants of order 80 and of 1 determinant of order 81.

1.5 ISOTACTIC CHAINS

Isotactic chains are chains of the (-CH2-CHR)„/2 type, in which two differ-ent bond-types occur in succession. The R-groups are positioned in such a way that each asymmetrical carbon atom has the same configuration. Lifson^'» has indicated how for these molecules the end-to-end distance can be calculated taking into account the rotations independently in pairs. It is assumed that rotations around the bonds on either side of the CHa-groups are dependent on each other because of interactions between the large R-groups, in contrast to the rotations of bonds on either side of the CHR-groups which are assumed to be independent of each other. Lifson's derivation leads to:

<rl>lnl' = [(1 + <T„>)(1 - <T„T„+i>)-»(! + <T„+,>)]3.3 (1.11) On the assumption that all interactions are independent, the above-mentioned equation can be simplified to:

<rl>lnP = 1-COS0. l - < c o s . p > ^ - < s i n ^ > ^ ^^ ^2) 1 -1- COSÖ (1 + <cos<j9>) + <sin^>

appUcabiUty of the above-mentioned formulae, however, is very limited, since in reality independent rotations occur seldom or never. In particular in isotactic polymers, which crystallize easily, large interactions must be expected. For this reason the extension made by Hoeve" to Lifson's cal-culations, is important. He allows that also the rotations of bonds on either side of the CHR-groups are dependent on each other. Hoeve's derivation differs little from that given in Section 1.3. It is, however, essential to define two w matrices (w' and w") and two F matrices (F' and F"), belonging to the bond types prime (') and double prime ("). The simplest formula obtained eventually is:

<rl>lnf = E3 + X*ZX + X*(D" + R)(E,„ - R)-'(E3„ + Z)X3,3(1.13) in which Z = X-' W D' W" and R = A"' W' D' W" D".

X, X* and X respectively represent the largest characteristic value of the matrix product W' W" and the correspondmg characteristic row and characteristic vector, after multiplication of then elements by Es.

Chapter II

CALCULATION OF THE END-TO-END DISTANCE FOR POLYETHYLENE CHAINS

2 . 1 INTRODUCTION

Application of the formulae for the calculation of the end-to-end distance of chain molecules, dealt with in Chapter I, requires extensive knowledge of the rotation potential round the chain bonds, of the bond lengths, of the magnitude of the bond angles and of the interactions which can occur be-tween neighbouring bonds. Thanks to the work of Mizushima'' and Pitzer'* on normal paraffins, the form of the rotation potential of polyethylene is known. There exist three minima in the potential energy plotted as a function of the angle of rotation round the C-C bond, one in the trans position, t, where ^ = 0°, and two symmetrically disposed gauche posi-tions, gL sod gR, for cp •= 120° and cp = 240° respectively. Of these three positions the trans position is the most favourable. With the aid of spectral analysis*' and calorimetric measurements*», the energy difference between the trans and gauche positions has been determined at 500 cal/mole. This model is also supported by electron diffraction studies. With the aid of this technique information can also be obtained about bond lengths and bond angles in lower n-alkanes. In gaseous n-butane Bartell*' found

1.533 A for the C-C bond length and 1.108 A for the C-H bond length. The bond angle $ is 112°. Measurements made on n-pentane, n-hexane and n-heptane show about the same values*^. The magnitude of the bond angle can also be calculated from X-ray examination of polyethylene crystals. The X-ray spacing along the chain axis in polyethylene crystals is 2.534 A*». By combining this value with the value of the bond length, a very accurate calculation of the bond angle can be made. Thus calculated, it is found to be 111°30'.

With the aid of the above-mentioned data, the end-to-end distance can be calculated according to the Eyring and Taylor formulae for independent rotations dealt with in Section 1.2. Eyring's formula, in which free rotation between the bonds is allowed, yields a value of 2.15 for the reduced mean square of the unperturbed end-to-end distance, whereas Taylor's formula, in which the existing rotation barrier is taken into account, yields a value of 3.53 when applied at 100° C. Taylor, however, already realized that calculations with the aid of formulae for independent rotations cannot be reUable, since it appears from model studies that the energy of a rotational

state is influenced by the states of its neighbours. Especially in confor-mations in which sequences of two gauche positions with opposite rotatio-nal angles occur, there are strong interactions.

Hoeve^» and Nagai^i made calculations with the aid of Equation (1.9) in which interactions between the nearest neighbours are taken into account. Both authors, however, did not use a priori estimates of the energy values of the various combinations of rotational states. Consequently, it was impossible for them to make independent calculations of the end-to-end distance. Instead they used the experimentally determined values of the end-to-end distance and its temperature dependence to obtain information about the energy of a certain combination of rotational states. For the energy of a sequence of two gauche states with opposite rotational angles Hoeve thus calculates a value of 2500 cal/mole. Nagai found the best agreement by attributing an infinite energy to this sequence and by putting the energy difference between the trans and gauche positions at 800 cal/mole. The reliabihty of these results, however, is very low, since this method presents the following difficulties.

1. In the calculations it had been assumed that interactions between neighbouring bonds occur only when two gauche positions with opposite rotational angles succeed each other. Examination of chain models has shown, however, that other combinations also lead to interactions.

2. The correctness of the experimental values used for the end-to-end distance and its temperature dependence is doubtful. Hoeve estimates the uncertainty in both values, determined by Flory, Ciferri and Chiang-^ and by Ciferri, Hoeve and Flory^^ at 10%. This, however, applies only to the uncertainty due to experimental errors. Imperfections in the theories used to derive the desired quantities from the experimental data, may lead to considerably larger errors. This appears clearly from Table 2.1, in which various experimentally determined values have been included.

<r2o>//i/- Authors Mode of determination 5,4±0,6 Kurata and Stockmayer^* h ] , M„, and K-S theory 6,5 ±0,7 Flory, Ciferri and Chiang-- M, M„,and Orofino theory

10,7 Krigbaum^» [vl M„, A^

21 Trementozzi^" Light scattering and A^ 17±3 Scholte and Koningsveld^' Light scattering in e-solvent

and M^-values

Table 2.1 Experimental values for the reduced unperturbed mean square end-to^ end distance of polyethylene chain molecules.

end-to-end distance, it is desirable to arrive at a reUable, independent cal-culation of this quantity. To that end an attempt has been made to calculate the energy values of various combinations of rotational states with the aid of functions which relate the geometry of the conformations to their energy. 2 . 2 FUNCTIONS RELATING GEOMETRIC VARIABLES TO ENERGIES

Many attempts have already been made with varying success to calculate the energies of molecular conformations. Interactions between non-bonded atoms, the bending of bond angles, changes in bond lengths and rotations round the bonds must be taken into account. In general these functions are obtained indirectly from spectroscopic or thermodynamic data.

2.2.1 Interactions between non-bonded atoms

The number of interactions occurring in a certain conformation between non-bonded atoms, and the intensity of these interactions, constitute an important factor in the calculation of the energy. It is, however, difficult to determine the exact function relating the distance of two non-bonded atoms with their interaction energy. Mason and Kreevoy^» assume that these interactions wiU approximate those occurring between free particles of an analogous form. Thus interactions between non-bonded halogen atoms are obtained from data about interactions between inert-gas atoms. Interactions between H-atoms can, however, hardly be compared with those between He-atoms, owing to the relatively large difference in nuclear charge between these two. A better agreement may be expected with the potential energy functions of the first electronically excited state of the hydrogen molecule, the 2' state.

This interaction function can be represented as follows: Er = K.<pir)

in which /sC is a constant, r being the distance between the atoms and <p(r) the potential energy function of the 2'-state of the hydrogen molecule. This function was calculated by Hirschfelder and Linnet^' from quantum-mechanical considerations and was put into an analytical form by Mason and Kreevoy by means of curve-fitting. In their calculations of rotational barriers. Mason and Kreevoy used the value K = I, since in this way the best agreement with experimental values is obtained. Wilson'», however, has shown that the formation of rotational barriers is not exclusively a result of interactions between non-bonded atoms, so that the good agree-ment is not a reason for maintaining the value K — I. From the theory concerning the valence-bond it may be deduced that for K a value of 0.5

is most probable'*. This value has been used by Pritchard and Summer'^ and by Pauncz and Ginsburg''.

Er = K3.716.10» exp(-3.071r)-89.52/r«]kcal/mole (r in A) The spherical symmetry shown by this function, which is evidenced by the lack of directional parameters and the neglect of the bonded-electrons effect, makes it doubtful whether it is suitable for calculating interactions between H-atoms separated by only one or two C-atoms. The interactions between H-atoms separated by one C-atom are, however, unimportant for the determination of energy differences between rotational states. Inter-actions between H-atoms, separated by two C-atoms, need also not be taken into account, because possible contributions of these interactions to energy differences can be more closely calculated with the aid of the torsional potential (see below). H-atoms, separated by several C-atoms, approach each other more 'head on', thus making the assumption of spherical symmetry more tenable and the neglect of the effect of the bonded-electrons more allowable.

In addition to H-H interactions, C-C and C-H interactions will also occur. Model studies have shown that for the calculation of energy differences only interactions between C-atoms, separated by two atoms, are impor-tant. In the conformations examined, the distance between such atoms is rather substantial. Moreover, there is partial screening by hydrogen atoms. With respect to C-H interactions, such remarks can also be made. In this case only interactions between atoms separated by three C-atoms are im-portant. Thus, it can be assumed that inclusion of the C-C and C-H inter-actions in the calculations is not essential. Moreover, in view of the great uncertainty in these fimctions, the introduction of these interactions could perhaps lead to bigger errors instead of to greater refinement. Neglecting these interactions is customary^s.'a.s».'''. IQ addition to Mason and Kree-voy, also Bartell'5 and Hendrickson'^ derived interaction functions. These functions consist of an attractive term as a result of the London dispersion forces and a repulsive term caused by the electron clouds. Application of these functions for the calculation of energy differences between confor-mations in linear alkanes, however, leads to considerable imderestimation of the energy.

2.2.2 Distortion of bond angles

When a valence angle is bent from its tetrahedral value (T = 109°28') to a value Ö, this bending will be accompanied by an increase in energy. It is assumed that the valence angle behaves like an ideal spring, to which

Hooke's law is appücable. The energy required for the deformation of the angle wiU then be proportional to the square of the displacement.

Eg = Kg(e- T)2 kcal/mole (angles in radians)

The values of the force-constant Kg can be calculated for various angles from the vibration frequencies measured in Raman and infra-red spectra. The following Kö values, taken from Westheimer's^e review, were used in

the calculations.

H-C-H Kg = 23.0 kcal/mole rad^ C-C-H Kg = 39.6 kcal/mole rad^ C-C-C Kg — 57.5 kcal/mole rad^

When one angle at a carbon atom is bent, the other angles will adapt themselves in such a way that the total energy of the six angles concerned is minimal. Hendrickson'* has calculated the most favourable energy states belonging to distorted C-C-C angles. For deviations up to about 10° there appears to be a linear relation between the various angles. The re-sults of his calculations are shown in Figure 2.1. In general, the energies

Figure 2.1 Angles around carbon: line mn is common to the perpendicular planes CCC and HCH. At minimum energy conformation the relation X^^ = x^ = 0.19o holds.

required for the bending of angles are not high. Thus in many cases energy is gained through bending since the gain in energy owing to re-duced interactions, exceeds the bending energy. In various preceding cal-culations^^, 33,34 the bending of valence angles has been incorrectly omitted. 2.2.3 Changes in bond lengths

Just as in the bending of angles, the energy increases with the elongation of bonds. The energy required for a change in length is again made pro-portional to the square of the displacement. The force-constants corres-ponding to changes in C-C and C-H lengths have been determined by

Herzbergs* and amount to 320 and 340 kcal/mole A^ respectively. From the magnitude of these values it appears that in n-alkanes the amount of energy required for a change in length is large compared with the gain in energy due to decreased interactions between non-bonded atoms. Conse-quently, the bond lengths may be considered constant and the values determined by Bartell*' may be adopted.

2.2.4 Rotations around the chain-bonds

It is impossible to give, with the aid of the functions discussed so far, a description of the shape of the potential curves which occur at rotations round single bonds. The reason is that the formation of the barriers is not exclusively caused by interactions between non-bonded atoms. Interactions between electrons associated with bonds, formed by the two connected atoms also play an important role". Quantitative calculation of these interactions is not possible, however. Therefore the potential energy curve generally accepted for the rotation in ethane^» must be introduced as a separate function in the calculation of energy differences between con-formations in n-alkanes.

Et = 1.4 (1 - cos 39?) kcal/mole

This function comprises both interactions between bonds and interactions between non-bonded H-atoms, separated by 2 C-atoms. Hendrickson'* and Bartell" also use this function in their calculations. Its neglect can lead to incorrect conclusions. The significance of this function is, however, not very great for the calculation of energy-differences between trans and gauche positions. Both in the trans and in the gauche positions (9? = 0 and cp = 120, 93 = 240), its contribution is in fact equal to zero. Only if deviations from these positions occur, the function will make a contribution.

2 . 3 POSSIBLE CHAIN CONFORMATIONS AND THEIR ENERGIES

With the aid of the formulae discussed in Section 2.2, an examination can be made into the most favourable conformations which can occur in poly-ethylene.

In the calculation of the distances between non-bonded atoms, use has been made of the geometric relations derived by Hendrickson'*. By means of these, distances between atoms separated by at most 7 atoms can be determined. The formulae have been programmed, so that the distances can be calculated with the aid of a computer (Telefunken TR4).

only three mmima in the rotation potential occur, viz. one in the trans position and two in the symmetrically disposed gauche positions.

Very recently similar investigations were published by Liquori et al.'' and by McCullough and McMahon^'. These authors reached the conclusion that minima occur not only in the trans and gauche positions, but also at angles of rotation of 90° and 270°. In their calculations, however, use was made exclusively of functions for interactions between non-bonded atoms. At rotational angles of 90° and 270°, however, the torsional potential makes a considerable contribution of 1400 cal/mole. Inclusion of this potential in the considerations makes these minima disappear, in accordance with the results of electron diffraction".

Thus, on application of Equation (1.9) for the end-to-end distance the energy of the following conformations only will have to be examined. 1. trans followed by trans (t-t)

a3 0 4 b1 b2 bS b6

H H H H

\ H H

H H

. / ' \ / ' \ '^'tX / ^ ' \

/

/H /W

/ \ / \

H " H " ^ H " H

Figure 2.2 The extended conformation of the polyethylene chain.

This energetically most favourable position is shown in Figure 2.2. For the determination of the minimum energy of this conformation, the dis-tances between hydrogen atoms on either side of the dashed line have been determined as functions of the magnitude of the bond angles. In the cal-culation it has been assumed that the bonds preceding tj and following tj ^ 1 also are in the trans position. This does not mean a restriction of the model, because possible extra energy caused by such a bond being in the gauche position, is taken into account when this particular position is con-sidered. The results of these calculations are graphically shown in Figure 2.3*. The energy minimum occurs at a bond angle of 111°45'. This value * For a specification of the energy values see Appendix 1.

3040 2 9 6 0 - E^, A 2880 -^ e -^'-^ o-^ 2aoo| |__ 1__ L. L_ i09^ n ó ' il? uP Ü3' Figure 2.3 The energy, E^, of the trans position as a function of the magnitude of the bond angle.

agrees very well with both the value of 111°30' determined by X-ray dif-fraction*» and with the value of 112° determined by electron diffraction*^. The C-C-H angles wiU then be 109°5'.

The energy of this position, Ett, amounts to 2 820 cal/mole*. Since this is the most favoiu-able position, it is taken as the zero position, so that ctt = 0.

In case of minor deviations of the rotation angle from the pure trans po-sition the higher energy due to the torsion potential is compensated by the lower interaction energy of the H-atoms separated by 3 C-atoms, which now can assume more favourable positions. See Figure 2.4*. The minimum in the potential energy curve round the trans position is con-sequently rather flat.

2. trans followed by gauche (r-gL and t-g^)

For the calculation of the end-to-end distance, the energy of this sequence can be put equal to that of the previous case etgi = etg-g^ = 0. All extra energy due to the gauche state can be taken into account in the following step. If one were to include this extra energy here too, some interactions would be counted twice.

3. gauche followed by trans (gi^-t and g^-t)

The energy of this state has been calculated as a function of various ro-• For a specification of the energy values see Appendix 1.

3600

Figure 2.4 The energy, E^, of the trans position as a function of the rotational angle.

tation and bond angles. From Figure 2.5 it can be seen that minima occur in the energy* if: a) the two bond angles öy and öj + i are deformed to

113°45', b) the gauche position lies at (p, — 120°, c) the C-C-H angles of two H-atoms, corresponding to as and bs in Figure 2.1, which interact strongly because they are located close to each other, are deformed to 111°, and d) the trans positions preceding and following the gauche position adjust themselves by rotating about 8°. The gauche positions are

'buffered', by the slight rotations of the trans positions. These deviations from the ideal trans positions are not unexpected. Measurements made by Bonham and Bartell*^ have shown that such deviations, even in butane, may occur.

In this state the energy difference egu(=eff-R^t) between the trans and gauche position is equal to 3340 - 2820 = 520 ^ 500 cal/mole*. This agrees closely with the experimentally determined values. This good agree-ment and the previously found agreeagree-ment between calculated and experi-mentally determined values of the bond angle in the trans position gives

lOOO 5 0 0 0 • > \

• ^

y

s^_^

Figure 2.5 The energy of a gauche position as a function of: a. the bond angles 6^ and 9^ ^ ^,

h. the rotational angle <pj,

c. the rotational angles <Pj_^ and <i>j^^,

d. the distorsion of two C-C-H angles (a3 and b3 of Figure 2.2).

Strong support to the reliabihty of the chosen energy functions and their appHcability to further calculations on the polyethylene chain.

In using other functions for the interaction between non-bonded atoms, this agreement is in fact far less good. With the aid of Hendrickson's for-mula'* no more than 25% of the experimentally determined energy dif-ference can be explained. McMahon and McCullough** calculate an energy difference of 1 000 cal/mole, whereas Liquori et al.^' arrive at a differ-ence of as much as 5 000 cal/mole.

4. gauche followed by gauche of equal type (gL"?L or gR-gK)

A minimum in the energy for two successive gauche positions arises when three bond angles are distorted to 113°45'. Since the energy required for the distortion of two angles is taken into account in the following step

(gij-t or gR-t), the distortion of only one bond angle needs to be considered here. With regard to the preceding case this yields an energy gain of 320 cal/mole*. In this case, however, the energy due to non-bonded inter-actions is found to be 530 cal/mole higher*. Because of this, the total energy of this position becomes 730 cal/mole higher than that of the trans position c^LffL = «pReR ^^ 700 cal/mole.

5. gauche followed by gauche of opposite type (gj^-gR or ^R-^L)

For this case it is always found that ESLSR ( = «»RffL) is larger than 5 000 cal/mole, even if the bond angles are distorted to 115° or more. An exact knowledge of this energy is not necessary since, in view of the accuracy of the calculations, the probability of this state can safely be put equal to zero, Taylor», BarteU*' and Nagai" had already assumed this on the basis of model studies.

In general, the second neighbour has no influence on the energy of a rotational state, so that em = en. There is, however, one hnportant ex-ception. When a gauche position is followed successively by a trans position and a gauche position of opposite type (gft-SK or gR-t-gi), the hemmed-in trans position cannot rotate either with the preceding or with the following gauche position. In this case the energy difference epL<pR ( = «ffR<ffL) with the complete trans position amounts to 1 300 cal/mole*.

Also when the gauche positions of opposite type are separated by several trans positions, this sequence remains unfavourable, since the rotation of the hemmed-in trans positions is hindered.

2.4 CALCULATIONS OF THE END-TO-END DISTANCE

After substitution of the e-values calculated in Section 2.3 in the W matrix of Equation (1.9) in which only nearest neighbour interactions are

• 7 0 •

» . I c°c) » - t (°c)

Figure 2.6 Values of <r^o>/n/2 and d In <r2o>dr, calculated according to Equation (1.9) as a function of temperature.

taken into account, the end-to-end distance <r2o>/nP and its tempera-ture dependence d In <_rh>/dT can be calculated. The results at various temperatures are graphically shown in Figure 2.6 and are mentioned in row 3 of Table 2.2. The calculations were carried out with a TR-4 elec-tronic computer. I II in IV V VI Calculated according to Free rotation

No correlation with neighbouring bonds First neighbouring bond correlation Second neighbouring bond correlation III -|- IV and partly the rotation of trans states V and estimated residual effects

<,r^a>/nP 100° 2.15 3.53 8.65 7.75 8.63 10±1 180° 2.15 3.21 8.37 7.32 8.17 -din <r''oy/dT 140° 0 -1.1810-= -0.7410-5 -0.71-10-5 -0.6810-5 -Table 2.2 Calculated values for the reduced unperturbed mean square end-to-end distance of polyethylene chain molecules.

The bending of the bond angles in the gauche position has been taken into account. However, it is not possible to accoimt for the deviations of the trans positions following or preceding the gauche positions with Equations (1.9), since in the diagonal matrix D only three transformation matrices occur, viz. for 0°, 120° and 240° respectively. Because of the extension of the calculation method given in Section 1.4 it is now possible at least partly to allow for this effect.

Apphcation of Equation (1.10) by means of which the influence of the fkst and the second neighbour bond on the end-to-end distance can be investigated, requires knowledge of 27 enk values, which must be sub-stituted in the W-matrix. Thanks to the calculations in the preceding paragraph these are all known. The results of the apphcation of Equation (1.10) are given in row 4 of Table 2.2. Compared with the preceding case, the end-to-end distance has substantially decreased. This decrease corres-ponds with the results of studies on short chain molecule models, which indicate that the sequences gj_-t-gR and g^-t-gj^ have a relatively large end-to-end distance. Putting an energy restriction of 1 300 cal/mole on these sequences limits then- occurrence, so that a decrease m the dimen-sions of the chain can be expected.

In these calculations the matrix D contains nine transformation matrices, three of which belong to the sequences t-t, t-gi^ and t-g^. By replacing the To" matrices belonging to t-gi^ and t-gR sequences by Ts" and T352", it is possible to include the effect of the rotation of the trans position on the

end-to-end distance. Row 4 of Table 2.1 shows that this leads to an mcrease in the dimensions. This increase again is corroborated by studies on mole-cule models.

Two effects, which influence the end-to-end distance now remain to be studied. Unfortunately, these effects cannot be taken into account quantita-tively. Sufficient data are available, however, to determine theh: influence with a reasonable degree of accuracy.

1. The rotation of a trans position following a gauche position

The effect of this rotation on the end-to-end distance cannot be calculated quantitatively since it is impossible to introduce in the D-matrix a dif-ference between transformation matrices of trans positions which are and which are not preceded by a gauche position. However, from model studies on short (5 bond) chains it follows that the rotation of a trans position, following a gauche position has the same effect on the end-to-end distance as the rotation of a trans position preceding a gauche position. Consequently, it seems logical to assume that both effects will also have the same influence on the end-to-end distance of a long chain. Thus, an increase in <^r^o^/nl^ of 0.9 units (rows 4 minus 5 in Table 2.2) can be expected.

2. Two gauche states of opposite type, separated by several trans positions

{gl^-t.. t-gR or gR-f. . t-gi).

Sequences of the type gi^-t-t-g^ are energetically hindered because neither hemmed-in trans positions can follow the rotation of the gauche positions. Again, model studies show that because of the energy restriction on these rather coiled sequences, the chain length will increase. It seems, however, quite unlikely that such an influence of the third neighbouring bond should exceed the influence of the second neighbouring bond, which is 0.9 units (rows 4 minus 3 in Table 2.2). Moreover, the effect is reduced by the mfluence of the gi^-t-t-t-g^ and g^-t-t-t-gj^ sequences, since the energetic hindermg of these positions leads to a decrease in the end-to-end distance. Therefore a positive correction of 0.5 units in <.r^o'>/nl'^ seems appro-priate. In Table 2.2, row 6, a fmal value of 8.6 -|- 0.9 -f- 0.5 = 10.0 is given for the reduced mean square end-to-end distance. Although this value is less exact than the other values, it should certainly be reliable within 10%. For d l n < r 2 o > / d r no separate value could be calculated, but this should not be a serious shortcoming, in view of the relatively minor influence of the other effects on this quantity.

2 . 5 FORM OF THE CHAIN

opposite types, even when separated by several trans positions, are improb-able. For a rough estimate let us assume that the probability of these sequences amounts approximately to:

^»^'^ = 0.085

^gttl + WgLtflL + ^gUgti + '^gLgU + ^gLgLgL + ^gLgRt + ^gLgf^gL + ^gLgRgR

Consequently, the chain can be considered as being built up of several parts, each containing 1/0.085 = 12 C-C bonds on the average, in which either exclusively gj^ and t or exclusively gR and t combinations occur. Very loosely one may speak of a kinked helical structure, with the imder-standing, however, that the helical parts are irregular, because the se-quences of gauche and trans positions are irregular.

Chapter HI

ISOTACTIC POLYPROPYLENE

3.1 CONFORMATIONS OF THE CHAIN

In Figure 3.1 an isotactic polypropylene chain in its fully extended form has been drawn. Each chain bond connects a -CH2- group with a -CHR-group. Looking from the -CH2- to the -CHR- group, the latter group can

C C

be in the Q or in the c configtiration. The relative bonds are H R R H

distinguished as prime (') and double prime (") bonds. It can easily be seen that in an isotactic polymer the sequence of these bonds is regular (') (") (') (") (') ("). The (') and (") bonds are mkror images of each other.

CH, CH, CH, CH,

V V V \/

A /^

A

H H H

Figure 3.1 Schematic diagram of an isotactic polypropylene chain in the extended conformation.

Consequently, a (') bond with a rotation angle 99 and a (") bond with a rotation angle (360° - cp) will have equal potential energies. With the aid of a mirror reflection with respect to a plane through a (") (') bond-pair and a second reflection with respect to a plane perpendicular to it, it may be seen that for each function of two angles, 991 and 992, belonging to a (") (') bond pak

f(<i»i, <P2) = f(360 - (P2,360 - (pi).

This relation, derived by Lifson'^, has been used in our study of the various possible chain conformations. For that study several potential curves

belonging to rotations round the chain bonds have been calculated. In these calculations use has been made of the energy functions dealt with in Chapter II. The magnitude of the bond angles has been determined with the aid of the known X-ray data on polypropylene crystals*^. The angles appear to have been bent to 114°.

Figure 3.2 shows potential curves belonging to rotations round a (") bond (991) and round a (') bond (992). Since in many cases the contribution of the torsional potential at deviations from the ideal trans and gauche

posi-lAZ.

xz

1 » ' » /

• CKcol/molt) N ' « /

' ' ^—^ • ^ ^

60* 120' l»0'

Figure 3.2 Potential curves belonging to rotations round a (") bond 0i and round a (') bond ifi^. The rotational states of the neighbouring bonds are indicated. tions is compensated by more favourable positions of the H-atoms, the bottoms of the potential troughs are rather flat. The shape of the rotational potential curves appears to be strongly dependent on the rotational states of the neighbouring bonds, indicating a strong correlation between the rotational positions. The rotational states of the neighbouring bonds are indicated in Figure 3.2. Many conformations cannot appear at all be-cause of these strong interactions. From Figure 3.2 it appears that only the following conformations can occur.

A B C D (1) (1) (1) (1) 240° 240° 0° 0° (2) (2) (2) (2) 0° 0° 120° 120° (3) (3) (3) (3) 240° 0° 240° 0° (4) (4) (4) (4) 0° 120° 0° 120° The conformations A and B correspond respectively to left-handed and right-handed helices. The hehces contain three monomer units per turn. X-ray analysis*^ shows that isotactic polypropylene crystals are indeed made up of such helices. This lends fiurther support to the quaUtative siderations of Bunn*^ and Natta**. These authors state that the chain

con-formations in the crystalline state are determined principally by intramole-cular forces. The conformations C and D respectively show the transitions from right-handed to left-handed, and from left-handed to right-handed helices. From these considerations it appears clearly that even in statisti-cally coiled molecules in solution short-range order will occur. This order corresponds with that occurring in the crystalline state. The chain will con-sist of sequences of left-handed and right-handed heUces. Since the bottoms of the potential troughs run rather flat, this order may in some degree be disturbed by motions within the troughs.

In thek recently published calculations, Bkshtein and Borisova*' as well as Corradini, Ganis and Ohverio*» also arrive at the conclusions that the chain consists of hehces. Moreover this chain concept is supported by the high resolution NMR results of Bovey*' and Shimanouchi*^ on isotactic polypropylene and its two-unit and three-unit model molecules.

3 . 2 CALCULATION OF THE END-TO-END DISTANCES

The reduced mean square end-to-end distance of the unperturbed isotactic polypropylene chain can be calculated with the aid of Equation (1.12), which was derived by Bkshtein and Ptitsyn in 1954. ReUable results must, however, not be expected from it, because the preceding section has shown the rotations are certainly not independent. The same goes for the apph-cation of Formula (1.11) derived by Lifson. Consequently, use will have to be made of Hoeve's Equation (1.13), in which the nearest neighbour interactions are taken into account. The apphcation of this formula, how-ever, requires precise knowledge of the relative energies of the possible sequences of rotational states. In this case it is hardly possible to perform a calculation of the minimum energy of these sequences with the aid of the functions of Chapter II, as reUably as was done for polyethylene, in view of the large number of interactions which occur and the numerous possibilities for angle distortion and for deviation from the ideal trans and gauche positions of the rotation angles. From the symmetry relations it can be seen, however, that the energy of the 0° - 120° - 0° - 120° sequence must be equal to that of the 240° - 0° - 240° - 0° sequence. These are the most favourable positions, so that e'tgi = s'aKt = 0 and e"g\^t = e" tg-p, = 0. Examination of the model further shows that the 0° - 0° sequence which transposes the right-handed into the left-handed heUx form, causes no extra interactions, so that e"tt — 0. Transition from the left-handed to the right-handed helix via the 120° - 240° sequence is energetically much more unfavourable, in view of the serious interactions occurring between H-atoms and CHs-groups, separated by 4 chain bonds.

Especially, for this position a reliable calculation of the energy is hardly practicable. Borisova and Bkshtein*^, however, have attempted to do this and have calculated an energy difference of 2-3 kcal/mole with the pre-ceding positions. In these calculations it has been assumed that six ro-tation angles can deviate 10° to 20° from the ideal positions. Moreover, opening of the bond angles up to 120° is allowed. The apphcability of the formulae to the calculation of the energy belonging to such large distortions is doubtful, however. Since a rough calculation with our energy functions lead to a transition energy of the same order of magnitude, the value of 2500 cal/mole for e"»L!7R of Borisova and Bkshtein has been used for further calculations in spite of the objections which one may have against thek calculation.

The energy of the remaining sequences of gauche and trans positions is so high that thek probabiUty can safely be put equal to zero.

The e' and e" values are substituted in the matrices W' and W" of Equation (1.13) respectively. The transformation matrices for 0°, 120° and 240° are substituted in the two diagonal matrices D' and D". Devia-tions from the ideal trans and gauche posiDevia-tions are consequently not taken into account. At 100°C a value of 53 is then obtained for the reduced mean square end-to-end distance. This value is very high compared with the experimentally determmed values (see Chapter IV). This is probably due to the fact that the great regularity in the isotactic molecule cannot be sufficiently described by Hoeve's equation in which only nearest neigh-bour interactions are taken into account. For this reason the following attempt has been made to find another method of calculation.

It is possible to calculate the average number of monomer units per heUx, p, with the aid of the energy data available. Let us call f{=\/p) the

frac-tion of the monomer units, which is involved in a transifrac-tion. At equili-brium the free energy of the chain as a function of ƒ will be a minimum:

^ = 0 (3.1 d/

The free energy of the chain can be represented as follows:

F = i/N(e'„ + £,;,,«) + RTNflnf+ RTN{1 - / ) l n ( l - ƒ ) (3.2) where N (= n/2) is the number of monomeric units.

A combination of Equations (3.1) and (3.2) yields the following expression for the averag number, p, of monomers per helix.

p = l-(-exp(e'„-t-e,i^R)/2Rr

Calculated values of p at different temperatures are listed in row 2 of Table 3.1.

Each monomer unit contains two C-C chain bonds, which form an angle of 114°. One of these bonds is parallel to the axis of the hehx to which it belongs. Thus the average length L of these helices amounts to:

L = p/{l -I- cos(180° - 114°)} = 1.41 p/

Examination of a chain model shows that the axes of two neighbouring hehces make angles of 114°. It is now possible to regard the hehces as new chain units. Thus chains consisting of n/2p units are obtained. On the assumption that all hehces are equal in length, the end-to-end distance can be calculated with the aid of Taylor's simplified Equation (1.2) smce the dkections in which the helices can take off at a junction are symmetri-cally disposed around the trans positions. Thus

2p<rl> _ l - c o s l l 4 ° _ ^ ^ , y nil 1 +cos 114°

For the reduced end-to-end distance it follows from this that:

^ 4 ^ = 2.35 p (3.3)

nl

Column 3 of Table 3.1 shows the values of the end-to-end distance which have been calculated at different temperatures, according to Equation (3.3). They seem to be more reasonable than the very high values obtained by means of Hoeve's equation.

t(°C) 20° 100» 180» P 9.6 6.4 5.0 Eq. (3.3) 22.5 15.0 11.8 </-2o>/«/2 Eq. (3.4) Eq. (1.20) 24.2 15.7 53 12.0 - d l n < r 2 o > / d r Eq. (3.3) Eq. (3.4) 4.010-5 4.310-*

Table 3.1 Calculated values for the reduced unperturbed mean square end-to-end distance of isotactic polypropylene chain molecules.

Moreover, our simple treatment agrees very well with the much more complicated matrix derivation of Corradini and Allegra (Column 4, Table

3.1). These authors again use the calculation methods developed by Lifson and Nagai. The transformation matrices are, however, combined in paks. Thek formula ultimately obtained reads:

<r^>/„/2 = l p ( l _ L ) (3.4) 3 2p

The values 4.0'10-3 and 4.3-lCM calculated for the temperature depen-dence of the end-to-end distance according to Equations (3.3) and (3.4) respectively, also agree very well. For these chains the temperature depen-dence is substantially larger than for polyethylene.

10 20 30 40 Figure 3.3 Plot of the end-to-end distance of isotactic polypropylene as a function of the transition energies. i

In Figure 3.3 the end-to-end distance, calculated with the aid of Equa-tion (3.3), has been plotted as a funcEqua-tion of the transiEqua-tion energies (e"^L»R -|- e"tt values). From this it appears that the magnitude of these energies has a considerable influence on the end-to-end distance. In view of the difficulties occurring in the calculation of these energy values, no great accuracy can be assigned to the calculated end-to-end distance. Allowing a

1000 cal/mole variation in the calculated energy, the values 15±5 and —(4.0±0.5)-10-3 for <r2o>/n/2 and d hi <rh>/dT are obtained at