Conformational properties of poly(alkene sulphone)s in solution: 2. A rotational isomeric state study of the poly(cyclohexene sulphone) chain

Conformational properties of poly(alkene sulphone)s in solution: 2. A rotational isomeric state study of the poly(cyclohexene sulphone) chain A . H . F a w c e t t a n d K . J. I v i n

Department of Chemistry, The Queen's University of Belfast, Belfast B T9 SAG, UK (Received 19 September 1974," revised 13 December 1974) The rotational isomeric states model, coupled with the Flory matrix method, was applied to the calculation of the unperturbed mean-square end-to-end distance in poly(cyclohexene sulphone) as a function of several parameters. The calculations were performed for atactic, isotactic and syndiotactic chains; the tacticity arises from the two possible ways, D and L, in which the rings can be attached to the main chain, assuming that the C - C bonds are all in the t conformation, as indicated by dielectric measurements. One of the three conformations about each C - S bond is strongly hindered and was given zero statistical weight whereas the other two were assigned weights determined by second-order effects, operating over three or more consecutive bonds, and dependent on the chirality of the rings and the coulombic interactions between adjacent dipoles. The calculations for the atactic model, in which a Monte Carlo procedure was used to generate the chain, were found to agree with experiment only if the parameter e was given a value of 5 or greater. This parameter is determined by the relative orientation of successive sulphone dipoles and was expected to have a value less than unity. It appears that specific solvation effects help to stabilize (t, g) and (g, t) C - S - C bond pairs relative to t, t. Other parameters are not so important. The calculations for the isotactic and syndiotactic models show interesting features, particularly for certain limiting conditions when cyclic or helical structures are generated. For these models there is a greater sensitivity to the various parameters than in the atactic case.

INTRODUCTION It was shown in Part 1 ~that neighbouring sulphone dipoles in poly(olefin sulphones) are likely to be correlated in orientation by dipole-dipole interactions. From the absence of a dielectric dispersion in the range 10 2 to 2 x 10 8 Hz in solutions of poly(cyclohexene sulphone) it was deduced that the sulphone groups attached to each cyclohexene ring take up a 1,2-diaxial configuration 2. There are therefore in effect only two axes of rotation between each pair of neighbouring dipoles, since of the three main-chain bonds between the sulphone groups one (the C - C bond) adopts solely the trans conformation with respect to the main chain. The simple matrix scheme of Flory 3, which allows the expression of second order Markov correlations of bond rotation, may thus be applied to this polymer to include explicitly the dipole-dipole interactions, without further elaboration to express third-order correlations 3. There are two distinguishable ways of substituting the chair form of the cyclohexane ring with 1,2-diaxial substituents. For this study we arbitrarily define the D and L forms as those depicted in Figure 1. The existence of these two chiral forms of the repeat unit allows the possibility of tacticity in the chain. Proceeding along such a chain the D and L units may be distributed randomly (atactic polymer), or they may occur alternately (syndiotactic polymer), or each unit may have the same chirality as the previous one (isotactic polymer). The results of the calculations will be independent of our arbitrary labelling of the chiral centres.

In this paper the influence of dipole-dipole correlation and tacticity upon the characteristic dimensions of the polymer is studied. The 13C-{1H} n.m.r, spectrum of poly(cyclohexene sulphone) made by the radical copolymerization of cyclohexene and sulphur dioxide shows two peaks of equal area for the main chain carbon 4, which indicates an atactic structure, as was also deduced for poly(propene sulphone) made in the same way s The properties of the rotational isomeric state model of the atactic chain considered in this paper may thus be compared with experimental data on the atactic polymer and an assessment

s%

D

Figure I

The two chiral forms of 1,2-diaxially substituted cyclohexane rings, L and D, as defined in the text

POLYMER, 1975, Vol 16, August

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A rotational isomeric state study ofpoly(cyclohexene sulphone): A. t4. Fawcett and K. J. Ivin

made of the reliability of the calculations of dipole-dipole correlations, made in Part 1. EXPERIMENTAL VALUES OF THE CHARACTERISTIC DIMENSION The characteristic dimensions of the polymer have been determined at 25°C in two solvent systems 2, although not under 0-conditions. Ko was obtained from viscosity measurements made in the good solvent benzene by the Stockmayer-Fixman 6 and the Inagaki-Suzuki-Kurata methods 7, and had a mean value of (5.4 + 0.4) x 10 -2 cm3/ g The Stockmayer-Fixman plot of the viscosity measurements obtained with the poor solvent system dioxane-cyclohexane (59:41% v/v) gave Ko = (5.56 +. 0.06) x 10 -2 cm3/ g. Since there is no rotation about the C - C main chain bond, (R2)/z, the mean-square end-to-end distance per repeat unit is a more significant quantity than the characteristic ratio ((R2)/n(12)av). It is obtained from:

o *F] 3/2 where @ is Flory's 0 constant 8, z is the number of repeat units in the chain, 3/0 is the weight of a mole of repeat units, q is a polydispersity factor 9 equal to (Mw/Mn) 1"5. • was given the value 2.6 x 1023 g - l , from theoretical x° and experimental determinations 11, and (Mw/Mn) was assumed the same as that (1.13 -+ 0.02) estimated for poly(hex-l-ene sulphone) 12 fractions obtained by similar fractionation techniques. The value of (R2)/z so obtained from the dioxane/cyclohexane results, 59 (+-2) x 10 .2o m 2 (repeat unit) -1 , was taken to represent the chain dimensions in benzene also.

ROTATIONAL ISOMERIC STATE CALCULATION Rotational isomeric state treatments of polymer chains relate a mean-square moment characteristic of a polymeric molecule to: (i) moments associated with individual bonds; (ii) the bond angles and dihedral angles of each rotational state; and (iii) statistical factors which express the relative probabilities of the various possible configurations at each bond. It is well established that these statistical factors should not only express the form of the rotational potential about each bond, but should also include the influence of the rotational states of the neighbouring bonds a. A method has been developed for performing such calculations in a convenient manner ts, which also allows the expression of tactic effects in the proper way. The present calculations were based upon equation (2), which is taken from the paper by Flory and Jemigan x4,

(M2)/n(m2)av=l

+

2 (Zn(~n2)av) Ijt00- • -0 IG~-I

duct), Gi is a function 14 of Ui, mi and T~($1)... Ti($r), ® indicates the direct matrix product, Ui = [Upq] i, the statistical weight matrix for bond i, Upq = exp{-(e(i p) + e(i,p ;i-1,q))/kT}, e(i,p) is the energy to promote the bond i to the pth state, e(i,p ;i- 1,q) is the energy of the interaction which occurs when the bond (i - 1) is then promoted to the qth state. T~(O,q~s)is the matrix to transform orthogonaUy from the coordinate system of bond (i + 1) to that of bond i, which has the rotational angle Ss. 0 is the angle between the two bond vectors, Z is the partition function J t U 1 U 2 . . . U n _ 1 J . The definition of the element Upq of the statistical weight matrix in terms of simple Boltzmann factors as carries the assumption, which may not be strictly valid, that the conformational entropy differences are zero. TRANSFORMATION MATRICES A segment of the main chain is shown in Figure 2. The three bonds S - C , C - C and C - S are labelled as vectors a, v and w, and are all shown in the trans conformation. In our calculations the virtual bond b, equal to v + w, replaces the two real bonds v and w ;this may be done because v is fixed in conformation. Bond lengths used were those employed in Part 1, and tetrahedral bond angles were assumed. The four right-handed coordinate systems shown in Figure 2 follow the usual conventions 3. Ta(X, ~ba) is simply given by: Ta(X, Sa) =

cosx sinx 0 sinxcos~a -cosxcosq~a sins a, sinxsin<#a -cosxsint~a - c o ~ a

but because w, the second axis of internal rotation in the repeat unit, does not coincide with the second bond vector b, the transformation matrix Tb(O~, ~b) is formulated as the product of three successive transformations: Tw(~k, (J~w), which has the same form as Ta(×, ~a), and transforms to system w; and two further transformations T(6) and TOO, respectively given by: T(8)

=

cos8 -sin6 on5 co~ 0

(4)

0 0

1

and

T00

=

,

0 0

o

(s)

ij

-1 0

-

Zo #

2

0

\1

,o, Ixo/°'

1(2)

IJOjm"l in which n is the number of bonds in the chain, M is the molecular moment, m is the bond moment vector, J is the r x 1 column matrix with all elements unity, and jt is its transpose, r is the number of rotational states for the bonds, G~I-1 is the product G1G 2 . . . Gn-1 (the serial pro-

574

POLYMER, 1975, Vol 16, August

(3)

Figure 2 Coordinate systems of the repeat unit of poly(cyclohexene sulphone)

A rotational isomeric state study ofpoly(cyclohexene sulphone): A. H. Fawcettand K. d. Ivin C

C

° S H

t

9+

js Figure 3

S

J

Rotational states t, g * and g - of the C - S bond w attached to an L-ring

which describe successive rotations to transform from system w to system b. Thus:

Tb(O,, ¢b) = Tw($, Sw)T(6)T(n)

(6)

(The subscript in 0¢, is used because the angle between bonds b and a' depends upon the conformation adopted by bond w, and thus upon Cw). STATISTICAL WEIGHT PARAMETERS

First order effects In considering first order effects in poly(cyclohexene sulphone) we have to deal with rotation about only one type of bond, namely the C - S bond. Conclusions concerning first order effects for the C - S bond may be applied equally to the neighbouring S - C bond. Figure 3 shows the rotational states of the C - S bond w attached to an L-ring. The t state is that in which the main-chain bonds are in the planar zig-zag configuration. In the t and g - states an oxygen atom is located above the ring, whereas in the g+ state a carbon atom, to which is attached a hydrogen and two other carbon atoms (part of the next ring, not shown), is in this position. In order to assess the relative importance of these three states we may compare the internuclear distances with the sums of the appropriate van der Waals radii. For the g+ state the carbon atom above the ring will be in close proximity to the two axial hydrogen atoms shown, which each form part of a methylene group. The distance between the carbon nucleus above the ring and the carbon nuclei to which these axial hydrogens are attached is 283 pro, which is appreciably less than twice the van der Waals radius of a methylene group (400 pm). In reality the sum of the van der Waals radii for the interacting groups will be substantially greater than this since the carbon atom above the ring is in fact part of a methine group in a substituted cyclohexane ring. Turning now to the t and g states the distance between the oxygen nucleus above the ring and the carbon nuclei of the ring methylene groups is 273 pm, which is again somewhat less than the sum of the individual van der Waals radii for a methylene group and

an oxygen atom (340 pm). However, in this case there is likely to be an attractive force arising from the dipole induced in the C - H bonds by the neighbouring S - O dipole, thereby decreasing the effective van der Waals radii. Such an effect appears to operate in poly(trimethylene oxide) where there is an attractive interaction x6 between oxygen atoms and methylene groups at a separation of 290 pm. Thus there are strong reasons for believing that the t and g - states will be greatly preferred over the g+ state. We therefore assign statistical weights of 1, 0 and 1 to the t, g+ and g - states respectively, for an L-ring. The corresponding weights for a D-ring are 1, 1 and 0 respectively. Second order effects, considered in the next section, modify this otherwise very simple situation.

Second order effects Here we have two things to consider. First, the effect of the rotational states of bond w on those of the subsequent bond a (the direction of the chain is defined in Figure 2); and second, the effect of the rotational states of bond a on those of the subsequent bond w. In both cases steric effects arise which depend on the chirality of the rings in the neighbourhood of the bond; but coulombic interaction between adjacent dipoles arises only in the second case, and is independent of the chirality of the rings. Correlation parameters for bond a. The interactions associated with the (w, a) bond pair depend on the chirality of the two rings adjacent to the bonds and on the conformations about each bond. There are four possibilities so far as the chirality is concerned, namely DD, LD, DL, LL, and three possibilities for the conformations about each of the two bonds, although, as discussed earlier, one of these for each bond may be assumed to have zero weight. In deciding what parameters to introduce at this stage we therefore need to examine 4 x 2 x 2 = 16 possible structures. Of these, four are depicted in Figure 4a and four in Figure 4b, the other eight being mirror images of these. It should be noted that sulphur atoms are attached to each ring either at SL or SD, the chirality of the ring then being denoted by the subscript. It will be seen that four structures summarized in Figure 4a have the same steric interactions but arise from four different combinations of configuration and conformation as listed in Table 1 ; likewise

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A rotational isomeric state study ofpoly(cyclohexene sulphone): A. H. Fawcett and K. J. Ivin

SD a

b

HL

S{)

/

SD

tl

H, I

Ftj

I:so

G

O

s[ Figure 4 Representation of eight of the sixteen accessible structures in relation to bond a. On each ring only one of the two positions labelled S D and S L is actually occupied by a sulphur atom; if the sulphur is attached at S D the ring has D-chirality, similarly for S L. (a) depicts f o u r of the combinations and (b) the other four as listed in Table I Table 1

Combinations of configuration and conformation cor-

responding to Figure 4

Figure 4a

Figure 4b

Configuration of ring attached to bond w

Conformation of bond w

Conformation of bond a

Configuration of ring attached to bond a

L' D D L L D D L

t g+ g+ t t g+ g+ t

t t gg+ g

D D L L D

g+t t

DL L

Table 2 The f o u r pairs of configurations/conformations corresponding to the structures in Figure 4b and their mirror images Configuration of ring

Confermation

Confermation

Configuration of ring

Combination

attached

of

of

attached

to bond w

bond w

bond a

t o bond a

1 2

L L

t g-

t g-

L L

1

3

L

t

g+

D

1

4 5 6

L D D

gg+ t

t t g-

D L L

1 1 1

7

D

g+

g+

D

a

8

D

t

t

D

1

Statisti-

cal weight

for the four structures summarized in Figure 4b. The eight mirror-image structures may be derived from those listed in Table I by exchanging L and D, and g+ and g - . For the structures shown in Figure 4a there are two possible interactions to consider, that between the two hydrogens labelled H i and that between the pair labelled HI. For

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a tetrahedral lattice the internuclear distance for each pair of atoms is 132 pro, which will give rise to a severe steric repulsion. From Hendrickson's data 17 we may estimate the repulsion energy at this distance to be in excess of 36 kJ/ mol. It is likely that the H/interaction will be relieved by a small twist from the tetrahedral lattice, but the Hi interaction will be less easily accommodated. For the structures shown in Figure 4b, there may be much smaller interactions between pairs of hydrogen atoms Hk and Ht, their internuclear distance on a tetrahedral lattice being 214 pm. To take account of this difference between the two groups of structures we introduce the pararneterf as the statistical weight of the structures shown in Figure 4a relative to those shown in Figure 4b. f i s expected to be very much smaller than 1. Let us now examine the more highly populated structures shown in Figure 4b and their four mirror images in more detail. They form four pairs in which the chirality of the rings is fixed but the two bonds w and a have a different sequence of conformatiolas. Combination 3 is the reverse of combination 5, which is the mirror image of combination 4. Combinations 3 and 4 may therefore be assigned equal weights of unity; likewise for combinations 5 and 6. However, combinations 1 and 2, and also 7 and 8 may have different weights and we introduce the parameter o as indicated in Table 2 to take account of this possibility. Examination of Figure 4b shows that in combination 7 alternate sulphone groups, labelled SD and SD, are brought closer together than in combination 1 (which is the mirror image of combination 8) where the alternate sulphone groups are labelled S L and S L. There is, however, no steric interaction and we may expect o to be close to unity. The statistical weight matrices for bond a are summarized in Table 3. If, as the above arguments suggest,f is zero, and o is unity it is apparent that (w, a) bond pairs are limited to very few possibilities, namely the eight shown in Table 2 which will be restricted still further if the chain is isotactic or syndiotactic. For example if successive rings have D chirality and bond w has t conformation then bond a

A rotational isomeric state study ofpoly(cyclohexene sulphone): A. H. Fawcett and K. J. Ivin must also have t conformation. This absolute determinance is an important feature of the model of which use is made in the next section. Correlation parameters for bond w. Here we consider how the rotational state of bond w is affected by that of bond a which precedes it in the chain, and which lies on the other side of a cyclohexane ring. Three possible sequences are shown in Figure 5. We shall not for the moment consider the dispositions o f the chain before Ca or beyond Cw, but simply recall that the conformations of the bonds immediately outside this segment are exactly determined as summarized in Table 2. The first point to note is that as the conformations in the (a, w) bond pair vary, so does the relative orientation of the two sulphone dipoles. Figure 5a shows the all-trans conformational sequence (on an L-ring), which is taken as the reference state so far as coulombic interactions of the sulphone dipoles are concerned. When either bond a or bond w is converted to t h e g - state, as for example in Figure 5b, the sulphone dipoles take up a relative orientation which we will define as an a-interaction, characterized by a statistical weight of a. When both bonds a and w are in t h e g - state, as in Figure 5c, there is a different relative orientation o f the dipoles, and we give this arrangement a statistical weight of % As previously concluded when examining first order effects,g+ states o f bonds attached Table 3

to an L-ring may be assigned a statistical weight of zero. For bonds attached to D-rings, it is the g - states of the bonds which have zero statistical weight and the a and 3' interactions then correspond to the (a, w) bond pairs (t,g +) or (g÷, t), and (g+, g÷) respectively. The calculations in Part 1 show that the excess energies associated with a and 3' (ea = -kTlnc~; e.r = -kTlnT) might be expected to be of the order of 5 and 9 kJ/mol respectively. This would suggest that 1 > a > 7. However, the assumption of a constant relative permittivity of the medium and the neglect of specific solvent effects could well upset this order and we have therefore not restricted the values of ea and e.r in the calculations. One further type o f interaction can be taken into account as a result of the severe restriction of available conformational sequences. If one examines sequences of three rings there are 8 possible combinations of chiralities coupled with four possible conformational sequences of the four bonds (two a and two w). Of these 32 arrangements, models show that in two (D g+g+D g+g+D and its mirror image) there is severe overlap at the ends of the sequence. Account o f this has been taken by introducing an additional parameter co in the statistical weight matrix for the w bond. The statistical weight matrices for bond w are summarized in Table 4.

Statistical weight matrices for bond a

THE COMPUTERPROGRAM

Bond a Chirality of ring adjacent to bond w

Chirality of ring adjacent to bond a L D Bond w

t

g+

g--

t

g+

g-

i o°' 0oo'1 li ,o oO°I i °oo o"1 i '°1oOOo

° o a

Equation (2) was computer-programed in Algol 60 to perform the calculations, (R2)/z being subsequently obtained using the relationship n(12)av/Z = a 2 + b 2 = 10.7 x 10 -20 m 2 (cf Figure 2). Table 5 contains the U matrices deduced for the rotatable bonds of the repeat units of the tactic polymers. The rotational states there shown with zero weights were excluded from the calculations, reducing the problem in effect to a 2-state model, with a consequently smaller requirement o f computer time. For the atactic chain the random sequence of ring chiralities was simulated by a Monte Carlo technique ~a, 400 or 800 chemical repeat

Cw

\s

o\ S

b

C W

13

Ca

(~)

Ca

Figure 5 An (a, w) bond pair on an L-ring showing three conformational sequences having different coulombic interactions bet~veen adjacent sulphone groups: (a) all-trans structure (the reference state); (b) bond a, g - ; bond w, t (an example of the e-interaction); (c) bonds a and w both g-- (the 3,-interaction)

POLYMER, 1975, Vol 16, August

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A rotational isomeric state study ofpoly(cyclohexene sulphone): A. H. Fawcett and K. J. Ivin Table 4

Statistical weight matrices for bond w

Bond w Chirality of central ring L

Bond a

triads (LLL, DDD)

g+

g-

g+ g-

0 0

g+ g-

0 0

Isotactic

t

D t

g+

g-

0 co3,

to3, 0

0 0

0 7

3, 0

0 0

,

triads (DLL, LLD, DLD, LDD, Other

DDL, LDL)

Table 5 U matrices for the isotactic and syndiotactic repeat units. The matrices refer to the bonds in heavier type.

Isotactic*

r

Syndiotactic*

f67

--S--(C)--C--(S)--

L

ifi0°

7 J

0 0 010 013"~

F

,"C7

--(S)--C--(C)

L

i i]00 0

v'~ s--(c)--c--(s)--

! i0 3, 0~0 i( Of 1

7 J

in the form of plots of (R2)/z against the appropriate e value in Figures 7a, 7b and 7c. It will be seen that in the regions close to f = 0 (high ca), w = 0 (high eo~), and o = 1 (e a = 0) there is no significant dependence of (R2)/z upon any of these parameters taken singly. Thus the most important parameter, so far as the atactic chain is concerned, is a and, unexpectedly this has to be assigned a value considerably greater than unity in order to obtain agreement with experiment. We must now consider what this means in physical terms and how it can be explained, a is the statistical weight of (a, w) bond pairs which have one t and one g conformation, relative to those in which both are t. This means that although in the polymer chain all the C - S bond dipoles cancel for both types of bond pair (since the C - C bonds are always t) the adjacent sulphone groups prefer to take up a non-parallel arrangement in benzene or in dioxane/cyclohexane mixtures despite the fact that simple coulombic considerations suggest that such an arrangement should have the higher e'nergy. The difference in coulombic energies is changed in magnitude but not in sign if a different value of the relative permittivity of the medium is used in the calculation. In order to account for a value of a greater than 1 it is necessary to postulate a differential solvation effect,

; ; i 200

* The main chain is shown as a planar zig-zagviewed from above, with the atoms in parenthesis being below the level of the adjacent atoms. The semi-circles represent the remainder of the 6-membered ring, the chirality being indicated as D or L.

units being generated. Matrix squaring was then used to find a limiting value of(R2)/z. The standard deviation of the mean characteristic dimensions of several such Monte Carlo chains was taken as an indication o f the reliability of the simulation process.

RESULTS OF CALCULATIONS AND DISCUSSION We give first the results for the atactic chain since here we can make a direct comparison with experiment. The results for the isotactic and syndiotactic chains show some interesting features which may have a wider bearing than on the particular polymer under consideration.

T~ t-

a

150

"6

im

~E I00 q: V _0 5G

I

-I0

A tactic chain A preliminary calculation was made using the following values, based on the earlier discussion: f = 0, o = I, co = 0, e~, = 5.1 kJ/mol (a --- 0.13 at 25°C), e7 = 9.0 kJ/mol (7 = 0.026). This gave (R2)/z = (211 -+ 33) x 10 -20 m 2 (repeat unit) -1 which is nearly four times the experimental value. Since the discrepancy is most likely to stem from an incorrect estimation of a and 3' (the coulombic parameters) these were varied first in order to discover what values were necessary to bring the calculated values into line with experiment. Figure 6a shows that there is marked dependence of (R2)/z on ea for all values of e7 used. The important thing to notice is that the calculated values of (R2)/z lie in the region of the experimental value only for values o f ea more negative than - 4 . 0 kJ/mol (a i> 5.0). The dependence on e.r in this region is only slight, as shown in Figure 6b. We have also tested the effect o f m a k i n g f a n d w finite, and of allowing o to depart from unity. The results are shown

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POLYMER,

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i

I

O

IO

c a (kJ/rnol)

T~oo c

b

"6 (3. E

I

50

A O

V

_o

0

-IO

O

IO E~, (k J/tool)

20

Figure 6

Atactic chain. (a) Dependence of (R2)/z on ee for values of e3, equal to 10.5 (O), 0 (A), and --10.5 (El) kJ/mol, and taking f = t o = O , o = l . (b) Dependence of (R)/z 2 on e~/taking ~a = --6.28 k J/tool, f = to = 0, o = 1. Lengths of vertical lines are equal to two standard deviations. - - - , denotes experimental value of

(R2)/z

A rotational isomeric state study ofpoly(cyclohexene sulphone): A. H. Fawcett and K. J. Ivin

ioo

a

14.j r"

0""¢~ o

O.

~

n

~

,:.

¢,

E 50

A V 0 o

0'

I 0' el

I00

2b

(kJlrnol)

b

t-

G}

o ~

-

-

o-- --~ -- - - - - --~'~--- --o-- - •

~50 o

A V

been estimated that the g form is stabilized by this effect to the extent of 1.7 kJ/mol. Other things being equal, this effect will be proportional to the square of the moment o f the dipole interacting with the solvent (regarding the effect as arising from a dipole-induced dipole attraction). The sulphone dipole is more than twice as big as the C-C1 and C - B r bond dipoles, so that an effect of the order of 8 kJ/mol in the present system is not unreasonable and will reconcile the value o f a deduced from the chain dimensions with that estimated from simple coulombic interactions. The presence o f the cyclohexane ring to one side of the chain contour may also be a factor assisting differential solvation of the (t, g) and (t, t) states. It may be noted that differential solvation is also believed to be an important factor in the interpretation of solvent effects on n.m.r. chemical shifts 2°, end-to-end distances 2x and mean-square dipole moments per repeat unit in polysulphones 22. In an earlier paper 2 we noted that the free rotation parameter, ((R2>/(R2))1/2, had a value of 1.24, (R 2> being derived on the basis'of two freely rotating C - S b6nds per repeat unit. It was concluded that the energy levels for the three conformations about the C - S bonds were close together. The present, more refined, treatment shows that this is not correct; the number of available conformational sequences is much more restricted and the fact that ((R2>/ (Rez))1/2 is close to unity is fortuitous. J

0

Isotactic chain 0

I

I

0

I0

I

20

Ere ( k J / m o l )

150

/

C

,-.-,

+

I00 ,,",t

-~'+

+ +.-.L..,_~

÷

__r=_4/-J

50 V o 0

0

'

6

-I0

' I0

E:o. (k J / t o o l )

Figure 7

Atactic chain. (a) Dependence of (R2)/z on ef, taking ~ = - 5 . 2 4 k J/tool, e7 = --2.1 k J/tool, o = 1, to = 0. (b) Dependence of /z on eoa, taking e¢~ = - 6 . 2 8 k J/tool, e7 = - 2 . 1 kJ/mol, o = 1, f = 0. (c) Dependence of /z

in which the solvent tends to stabilize the (t, g) or (g, t) bond pair relative to the t, t bond pair. Such an effect has indeed been observed with small molecules such as 1,2-dichloroethane la where both benzene and dioxane have been found to solvate the gauche form preferentially; likewise for 1,2dibromoethane in benzene 19. In the second case it has

In the calculations for the isotactic chain the parameters T and 60 always occur as a product (see Table 5) so that we may treat 760 as a single variable. Plots of(R2)/z against e~, eo and (e.r + e~o) are shown in Figures 8a, 8b and 8c respectively. Comparison with Figures 6 and 7 show that there is a marked difference in the nature of the curves for the atactic and isotactic chains. For the isotactic chains the calculated characteristic ratio is very sensitive not only to the value oft~ but also to the values of o and T6o. However, it may be noted that the minima of many of the curves lie close to the experimental value for the atactic chain. The much greater dependence of
Syndiotactic chain For this model, two parameters, ct and 7, have been varied, f b e i n g taken as zero (see Table 5). The calculated values of 7 = 0. This corresponds to the right-hand side of Figure 9b. Under these conditions, the four rotatable bonds in the repeat unit have the sequence (g-, t, g÷, t), in the limit of large z. The chemical repeat unit then has two bonds in the trans conformation, and the third in a gauche conformation. Gauche conformations along the chain have alternate sign, and the chain describes a ring. This is why the limiting value of(R2)/z is zero for all finite values o f a .

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1 9 7 5 , V o l 16, A u g u s t

579

A rotational isomeric state study ofpoly(cyclohexene sulphone): A. 14. Fawcett and' K. J. lvin (2). 7 > ot = 0. This corresponds to the right-hand side of

a

600

Figure 9a. Two sequences of conformations for the repeat B

,_~ ~ ~4OO o. e ^ ~200

O

J

O

I

-5

O

5

e ct (k J/tool) b

A

unit are equally probable under these circumstances. These are (g-, g - , t, t) and (t, t, g+, g+), and their repetition generates respective ly ri gh t-handed and left-handed helices, for which (RZ)/z ~ z and has no limit. The two helices cease to be mutually exclusive if the condition a = 0 is relaxed. As a increases relative to 7, the points along t h e chain at which transitions between the two helix forms occur come closer and closer together, until eventually the random coil form of the chain: is obtained. The convergence of(R2)/z to the limiting value in ~, variety of polymers has been found to be from below 14'ra. With certain values of a and 7, when conditions ase c~ose to those which lead to the chain describing a ring, it was found that the limiting value of (R2>/z was approached from above. This type of convergence arises from the quasiring structure of the polymer. A consequence of the form of the statistical weight matrices is that a set o f values of ca and e~ exist for each value of(R2)/z. It was found that for each set there was relationship between ea and e~ of the form e~ = 2ca + constant.

7~600

=

CONCLUSIONS The main conclusion from this work is that, for the atactic polymer, the parameter a unexpectedly has a value greater

~'E'400

A

a 200

200

i-

I

c

600

1

-~o

o e:o (k J/tool)

o

1

¢.~

io

/

~V 0

¢o. (kJ/rnol)

v

oo

;/\

\

'~100

EW

Figure 8

(R2)/z on e a for values of (e~ + e~) of (A) --6.3, (B) --2.1, (C) 8.4 k J/reel; with o = 1, f = 0. (b) Dependence of (R2)/z on e o for values of (A) e,), + e ~ = Isotactic chain.

(a) D e p e n d e n c e of

o

j

,o

2'0

3'0

Ey (k J/tool)

ea = 0, (B) e3, + eta = --e a = 4.2, (C) e.y + eta = ea = - 4 . 2 , (D) e3, + eo~" = c a = - - 8 . 4 k J/tool; with o = 1, f = 0. (c) Dependence of on (e 3, + eco) f o r values of e a of ( A ) - - 8 . 4 , (B) - 4 . 2 , (C) 2.1 k J/tool; w i t h a = 1, f = 0

(RL)/Z

580

POLYMER,

1975,

Vol

16, August

Figure 9 Svndiotactic chain. (a) Dependence of (R2)/z on ea for values of e 3, equal to ( A ) 10.5, (B) 21 k J / m o l , with f = 0 . (b) Dependence of (R2)/z on e,), for e a = 6 . 3 k J/tool, f = 0

A rotational isomeric state study ofpoly(cyclohexene sulphone): A. H. Fawcett and K. J. Ivin than unity, perhaps even bigger than 5, whereas a simple treatment in terms of coulombic interactions between adjacent dipoles would have suggested a value less than unity. It thus appears that specific solvation effects help to stabilize (t,g) and (g, t) states for C - S - C bond pairs in poly(cyclohexene sulphone). For the isotactic and syndiotactic models the value of (R2)/z is much more sensitive to the various parameters than in the atactic case and a marked dependence of the solution properties of such pol,'ymers on solvent is to be expected. ACKNOW LEDG EMENT A. H. F. thanks the Inter-University Council for the award of a Fellowship.

6 7 8 9

1020

10 11 12 13 14 15 16 17 18 19 20

REFERENCES 1 2 3 4 5

Fawcett, A. H. and Ivin, K. J.Polymer 1975, 16, 569 Fawcett, A. H. and Ivin, K. J. Polymer 1972, 13,439 Flory, P. J. 'Statistical Mechanicsof Chain Molecules', lnterscience, John Wiley, London, 1969 lvin, K. J. and Stewart, C. D. unpublished results lvin, K. J. and Navr~[til,M. J. Polym. Sci. (14-1) 1970, 8, 3373

Stockmayer, W. H. and Fixman, M. J. Polym. Sci. (C) 1963, 1,137 lnagaki, H., Suzuki, H. and Kurata, M. J. Polym. ScL (C) 1966, 15,409 Flory, P. J. "Principlesof Polymer Chemistry', Cornell University Press, Ithaca, 1953 Ptitsyn, O. B. and Eizner, Yu. E. Zh. Techn. Fiz. 1958, 29,

21 22 23

Pyun, C.W. andFixman, M.J. Chem. Phys. 1965,42,3838 Berry,G. J. Chem. Phys. 1966, 44, 4450 Bates,T. W. PhD Thesis Leeds University (1965) Flory, P.J. Proc. Nat. Acad. ScL US1964,51,1060 Flory, P. J. and Jernigan, R. L. J. Chem. Phys, 1965, 42, 1965 Hoeve,C. A. J. J. Chem. Phys. 1960, 32, 888 Mark,J.E.J. Polym. Sci.(B) 1966,4,825 Hendrickson, J. B. J. Am. Chem. Soc. 1961, 83, 4537 Oi, N. and Coetzee, J. F. J. Am. Chem. Soc. 1969, 91, 2478 Abraham, R. J., Cavalli, L. and Pachler, K, G. R. MoL Phys. 1966, 11,471 Fawcett, A. H. PhD Thesis The Queen's University of Belfast (1968); lvin, K. J. and Navvltil, M. Prepr. IUPAC Int. Syrup. Macromolecules, Boston, 1971, p 755 Ivin, K. J., Ende, H. A. and Meyerhoff, G. Polymer 1962, 3, 129; Bates, T. W., Biggins,J. and Ivin, K. J. MakromoL Chem. 1965, 87, 180 Bates,T. W., Ivin, K. J. and Williams,G. Trans. Faraday Soc. 1967,63, 1976 Mark,J. E. and Flory, P. J. J. Am. Chem. Soc. 1966, 88, 3702; Brant, D. A. and Flory, P. J. J. Am. Chem. Soc. 1965, 87, 2788

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581