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Elastic modulus of linear polymer crystals
Elastic modulus of linear polymer crystals T. R. Manley and C. G. Martin Department of Materials Science, Newcastle upon Tyne Polytechnic, Newcastle upon Tyne NE1 8ST, UK (Received 19 March 1973) A method based on energy balance over the repeat unit is used for the calculation of the elastic modulus of linear polymer crystals. The result for polyethylene agrees with that of
Treloar who used a statical analysis over the whole molecule. Both methods are much simpler than those of Shimanouchi and Miyazawa, the latter being based on the complete potential function for the polymer. Using force constants based on a Urey Bradley force field, the following values of modulus were found (all syndiotactic except PE) in GN/m2: PE, 299; PVC, 153; PVF, 212; PVA, 142; PAN, 236; PMMA, 63.
INTRODUCTION The method of Jaswon et al. 1 for the calculation of the elastic moduli of cellulose was modified and applied to the helical polymer crystals of poly(phosphonitrilic chloride) 2 and values for Young's modulus of 0"138 GN/m e and 1.66 GN/m ~ obtained for the uniform helix and cis-planar structures respectively. These results are similar to those calculated for other helical polymers, but as all experiments with (NPCI2)n used amorphous rubber, no direct comparison is possible. The method is equally applicable to linear polymers and it was therefore decided to apply it to crystals of polyethylene and the syndiotactic forms of poly(vinyl chloride) (PVC), poly(vinyl fluoride) (PVF), poly(vinyl alcohol) (PVA), polyacrylonitrile (PAN) and poly(methyl methacrylate) (PMMA). Different solutions have been proposed for both polyethylene3-5 and PVC 6. Treloar 4 performed a statical analysis using simple valence force constants. On the other hand, the Japanese workers 3, 5,6 calculated the modulus directly from the complete potential function for the molecule. We assume that the polymer chain in the crystal is isolated and infinite; the repeat unit alone is characterized. The problem of a free or fixed end group does not arise. The bond lengths and bond angles of the repeating unit are taken as degrees of freedom which can be varied independently of one another. This simplifies the choice of force constants since we derive the extensions due to bond stretching separately from those due to angular deformation. The force constants are selected from a Urey Bradley force field (UBFF), ih which only the 'pure' stretching or bending constants for the appropriate bond are taken. The mixed terms, i.e. bending-stretching interaction constants are meaningless in our method of modulus calculation. In obtaining these force constants we do, however, consider the effect of adjacent nonbonded atoms on the particular stretching or bending force constant in question. The number of force constants required for our analysis is a minimum and they a r e calculated from a knowledge of K, H and F, the stretching, bending and repulsive force constants for the
appropriate bonds or angles, which may be transferred from smaller molecules 7. The methods of Shimanouchi et al. 3, 6 and Miyazawa5 require the setting up of the complete Urey Bradley force field for the polymer repeat unit including both stretchingstretching and stretching-bending interaction terms. The relationships between internal parameters and the helical parameters of the particular polymer structure are also required. Some knowledge and expertise in spectroscopic techniques are required in order to set up the potential energy equation, remove the redundancies from it and derive the relationships between the helical and internal parameters. The calculation of the modulus requires the differentiation of all these equations and the solution of the resultant simultaneous equations. The present method, therefore, is much simpler. This advantage is particularly important with polymers more complex than polyethylene. The Japanese workers 5, ~ have treated polymers with up to four different chain units, e.g. -(-A1--A2--A3--A4-)-n but excluding -(-A1--A2--[Aa]x-)-n. On the other hand, the present method is readily applicable to polymers such as:
-(-A1--A2--[Aa]x--A z--A1-- [A3]y~)-n POLYETHYLENE The polyethylene molecule has an extended planar zig-zag structure in which the repeating unit is the --CH~-- group. This is characterized by the bond length r and bond angle ¢ (see Figure 1). The translational repeat unit is defined as SS' and is made up from ½rl, ~, r2, ¢, ½rl, (Figure 2). It is more convenient, however, to base it on C1C'1 bearing in mind that ¢ occurs twice. Both definitions give the same equations but the latter is more convenient. A set of rectangular Cartesian axes related to these quantities can be superimposed by choosing one of the carbon atoms as an origin and identifying one of the bonds as the direction of the x axis as shown in Figure 2. The y axis is then perpendicular to C1C2 and is in the
POLYMER, 1973, Vol 14, October 491
Elastic modulus of linear polymer crystals: T. R. Manley and C. G. Martin I
l
Hn-2 Hn-2
Now from equation (1)
l
Hn Hn
Hn+2.Hn÷2
•
-dO
i.e.
dLa = ~/2[sin0, - cos0]d¢ dLa = (cosf~sin0- sint2 cos 0)12d¢
•
(7)
Then utilizing the Jaswon et al. formulai:
Hn-i Hn-i
Hn÷lHn+l
C i ~ / O L \ 2 1 02W - =~) /O(dl,)2
Figure 1 Structure of polyethylene molecule
Y
the overall force constant C for the repeat unit can be obtained from: C-i=
S
(dLi) 2 , (dL2) 2 , (dLs) 2 K -t'~-t 2H
(9)
where K and H are the bond stretching and bond angle deformation force constants for - C H z - C H s - and -CH2-CH2-CH2- respectively. The factor 2H arises since the degree of freedom ff occurs twice in the repeat distance, (cf. SS' in Figure 2). The overall force constant C is then related to the elastic constant csz by the energy balance:
-"-X
C~
(8)
C2
/ Figure 2 Translational repeat unit of PE
½cs2e~V= W= ½C(dL) 2= CLSe~
(lO)
plane of the molecule. The z axis is out of the paper but is not required for this linear planar molecule. The auxiliary angle 0 is the angle between bond C2C~ and the x axis and is related to ff by" ¢=,~-0 (1) Vector components can now be assigned to the bonds in the repeat unit: CiC2 = (1, 0)ri
where A is the effective cross-sectional area of the repeating unit. The force constants used were taken from a UBFF, the deviation of which is shown in the Appendix and are in the form:
C2C ~= (cos0, sin0)r2
K = 1£2= Kcc + 2s~Fcc + 4s~Frtc
•
CL2 CL cs2= V = A
(11)
and
The repeat trait vector is given as: C1C~= g. L = (cosf2, sinf~)L
L = (r~ + r ~ - 2fir s cos~) i/9'
(3)
When the chain is deformed by a uniform strain along its axis the symmetry is still preserved and hence the three quantities ri, rs and ¢, still describe the chain configuration. These can be considered as three degrees of freedom which can be varied independently of one another and will be designated as h using our previous notation s. If we consider the repeating unit L to undergo a change in length dL, this change is made up of the sum of the small changes due to a small change in each degree of freedom, i.e. dL =~dL~; i = 1, 3 (4) These small changes are given as follows: dLi = ~(1, 0)dli = cost2dli
H=b2Hz
(2)
where f~ is the angle between the fibre direction along CiC'i and the x axis and L is the repeat distance given by:
where
1-I2= Hccc + t~Fcc + 3K/(81/SbS)
= (cosf~,cos0 + sinf~, sin0)dls
SYNDIOTACTIC PVC Syndiotactic PVC has an extended planar zig-zag structure as shown in Figure 3 and the modulus may be calculated in the manner used for PE.
(5) (6)
and dLa = ~/2:¢[cosO, sin0]d4, dO
(I) CH2 Figure 3 Structure of s-PVC
492 POLYMER, 1973, Vol 14, October
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Force constant data were taken from refs 3 and 8, since no complete set of UBFF constants was available for the polyethylene molecule. The molecular parameters are given in Table 1 and the values used in the force constant calculations in Table 2. The modulus (c22) for polyethylene is calculated as: 299 GN/m 2.
dLz = a(cos0, sin0)dl2
dO
(12)
qaa
(I) CH2
Elastic modulus of linear polymer crystals: T. R. Manley and C. G. Martin Table 1 Molceular parameters used in modulus calculations Tetrahedral bond angles are used throughout (109° 28') C-C bond distances are all 0.154 nm
Polymer (CH=CXY)n
X
Y
R(C-X) (nm)
R(C-Y) (nm)
Crosssectional area,* A (nm 2)
PEa s-PVCb s-PVFc s-PVAc s-PANc s-PMMAc
H CI F OH CN COOCH3
H H H H H CHa
0.109 0.177 0.135 0"413 0"216 0"154
0'109 0.109 0.109 0-109 0-109 0.154
0.1824 0"2746 0.2121 0-2152 0-3111 0'5565
Bond distances
Nature of
Repeat distance L" (nm) 0.2515 0-2515 0'2520 0.2520 0-2550 0.2510
* From unit cell measurements except for PMMA a Ref. 12 b Ref. 6 c Alexander, L. E. 'X-ray diffractions in polymer science'. Wiley, N.Y. 1969, p. 473 Table 2 Force constant data used in modulus calculations Nature of Polymer (CH¢CXY)n PEa,b s-PVCc s-PVFc s-PVAd s-PANe s-PMMAf
X
KC-C (N/m)
Y
FCH2-CHz FCXY-CXY (N/m) (N/m)
FCX (N/m)
FCy (N/m)
Hi
H2
(N/m)
(N/m)
(aNm) 0'2 0"23* 0-25* 0'17 Kz=0"01 ~=0"11 0'1
H
H
CI F OH CN
H H H H
280 340 340 280 200
96 33 33 33 38
96 96 150" 30 44
40 60 130 28d 44
40 40 40 40 54
11 11 11 20* 34
11 20 25* 11 27
COOCH3
CHz
340
96
33
83
40
11
20
* Estimated value a Ref. 3 b Ref. 8 c Ref. 6 d From propyl alcohol e Ref. 11 f From polyisobutylene and polypropyleneS
or in this case ,
OS ~, CH(2) J r 2 1
L
L
00 = (cosfl, sinfl) 2 = ~ 2 = ~L
) .'.
.~Ir12 t-u
,
dL1 = costld6
(14)
dL2 = ~(cos01, sin01)dl2 = (cost) cos01 + singl sin01)d/2
O
m
X
/ Figure 4 Translational repeat unit for s-PVC
The rectangular Cartesian axes are superimposed as for PE with a C atom as origin and a bond r21 as the x axis as shown in Figure 4. Although there are four C atoms in the repeat unit we are only considering deformation of the carbon skeleton and since ¢1=¢2 the alternate nature of the chlorine atoms along the chain will be mechanically indeterminate• Hence we need only consider half of the actual repeat unit with four degrees of freedom to be varied rzl, rlz, ¢1 and ¢2 and defining the auxiliary angles 01 and 02. The mechanical model is, therefore, almost exactly the same as for polyethylene and gives the following expressions for the small changes dL~.
(15)
which are exactly the same as in polyethylene. The angles ¢1, i.e,
,"
d01 = - d¢1 r12~
dLa= 24,~w
~-~
sin01-sint) cos01)d¢l
(16)
and also d_ _r12 0 La=n 2 0¢~ [-c°s0z'-sin0z]dCz -ra2rt - sin0z, cos0z]dCz dL4 = n~-
dLi = ~(1, O)dh where ~ is the direction cosine of the repeat distance vector Off' = (cosQ, sinfl)L = ri. L
since
d¢2 = - d02
.'.
dL4= r1,,2[- cost) sin0,~ +sinfl cos02]dCz z
POLYMER, 1973, Vol 14, October
(17)
493
Elastic modulus o[ /inear polymer crystals: T, R. Manley and C. G. Martin The overall force constant C for the repeat unit can then be obtained from:
Table 3
Calculated moduli of several polymer crystals
Calculated results
C-1
(GN/m~)
(dL1) 2 , (dL2) 2 . (dLs)2~_(dL4) 2 =
K
-
(18)
where K and H~x, H~ are the stretching and bending force constants for -CHCI-CHz--, - C H C I - C H 2 - C H C I - and - C H z - C H C I - C H 2 - respectively. DERIVATION OF THE FORCE CONSTANTS K, H~, H~ The Urey Bradley force field (UBFF) used for PVC is similar to that for polyethylene given in the Appendix but contains more terms due to the difference between CH2 and CHC1, i.e. the term in ARn in polyethylene ( C n - C n + l ) becomes a term in Ar12 and Ar21. Similarly the term in Affn becomes a term in A9~1and A¢2. Thus the required force constants are: K = K rl, = K r- = Kcc + S~fCa,CH, + s~2FcHelCHCt +
s~2Fccl+ 3s~zFcrI (19) and
H~ = p2H1 + t22p2FcHc1CHC1+ p 2 3 x / ( 8 l12r21r12)
(20)
H~=p2H2+t~1p2FcH,CH=+p23K/(81/2r12r21)
(21)
p
q
PE
999 189
s-PVC s-PVF s-PVA
153 219 142 236 63
340a 182b 290c 160d
s-PAN s-PMMA
Experimental results 9 (GN/m 2) 235
251
a Ref. 3 b Ref. 4 c Ref. 5 dRef. 6 pThis work q Other authors
where Z is the number of monomer units per unit cell (in this case Z = 4), M = molecular weight of a monomer unit (100.11), V= volume of the unit cell, N = Avogadro's n u m b e r = 6.023 x 1023 mo1-1, and the chain length L = 5.02014 x 10 -8 cm. The only X-ray density available for s-PMMA was 1.19 g/cm a obtained from the amorphous region. The crystalline density will be slightly higher but for the approximate nature of this calculation for cross-sectional area the value was thought to be adequate. The crosssectional area of the 'unit cell' was thus calculated as 111.291 × 10-16 cm 2 giving an effective cross-sectional area of chain as 55"645 x 10-16 cm 2. Substituting the above in equations (14) to (22) gave the modulus of s-PMMA as c22 = 63 G N / m L This low value of the modulus is attributed to the large cross-sectional area of the c h a i n = 5 5 x 10-16 cm 2 compared with 18.24x 10 -16 cm 2 for polyethylene, 27.46 x 10-16 cm 2 for PVC, etc., which is caused by the bulky methyl and ester groups in the side chain.
where sn = (r12 - r21cos¢2)/qn
s 2z = (r21- r12cos¢l ) / q22 s12 = (r12- r~cos¢')/q12 S21 = (r21 - -
Polymer
r~cos¢~)/q~t
tn = psin$2/qn t22 = psinfl/q22 and
p=(r12r21) 1/2 2_ 2 2-2r12r2tcos$2 qll--rl2+r21 q 22.9= r ~1+ r ~2- 2r21rl2cos$1 12 2 t2 ~ t .t q 12 = r 12 + r 2 - - z r 1 2 r 2 c ° s 9 2 '2
2
'2
q21= r 21+ rl
EXTENSIBILITY
-- 2r21r l' C O S ~ I
Using the values in Tables I and 2 the elastic constant c22 of s-PVC was calculated as c22 = 153 G N / m z. The simplicity of this method for calculating the elastic moduli of planar zig-zag polymer chains of type (A1-A2-A1-A2) enabled us to calculate the moduli of other linear polymers, e.g. PVF, PVA and PAN, by substituting data into equations (14) to (21) as for the s-PVC calculation. The parameters from Tables 1 and 2 gave the results shown in Table 3.
The values of modulus for polymers of type (A1-A2)n in Table 3 were used to calculate the extensibility o r f v a l u e . The extensibility is the force required to stretch a molecule by 1 ~o in the direction of the molecular axis and may be calculated from the modulus and cross-sectional area of a single chain 9. The f value is mainly dependent on conformation and for planar zig-zag type polymers usually falls in the range 0.4--0.5 nN. The f values for each of the polymers are shown in Table 4. DISCUSSION
SYNDIOTACTIC
POLY(METHYL
METHACRYLATE)
In this case very little X-ray information about the unit cell is available so the cross-sectional area of the unit cell was calculated from the X-ray density-volume relation:
ZxM
density= Vx N
494 POLYMER, 1973, Vol 14, October
(22)
The value for polyethylene obtained by the present method is identical to that of Treloar 4 if the same force constants are employed (Table 3). The improved values for the force constants in the present paper give a result lower than that of Shimanouchi 3 but close to that of Miyazawa 5. A much better result is expected when force constant values from a single source are available. As
Elastic modulus of linear polymer crystals: T. R. Manley and C. G. Martin Table 4
Extensibilities (fvalues) of several polymer crystals Modulus
Cross-sectional area
Polymer
(GN/m 2)
(nm 2)
/value (nN)
PE s-PVC s-PVF s-PVA s-PAN s-PMMA
299 153 212 143 236 63
0" 1824 0"2746 0.2121 0.2152 0.3111 0-556
0' 545 0-420 0.449 0.31 0"73 0"35
expected, the theoretical value of the elastic modulus in each case is higher than the experimental value obtained using X-ray methods 9. The present method is preferred because it is simplest. For s-PVC, this method gives a value close to that calculated by Asahina and Enomoto 6 (see Table 3). No X-ray data were available for PVC, but the f value obtained is within the expected range (see Table 4). Similarly, no experimental or calculated data are available for s-PVF but the extensibility falls within the expected range (see Table 4). Syndiotactic PVA gave a low result when compared to the experimental result (see Table 3). This could be due to lack of suitable force constants for substitution into the UBFF, but is more likely to be due to neglect of secondary bonding effects. We consider an isolated polymer chain, i.e. neglecting intermolecular forces. These are normally very small when compared with the primary valence forces along the chain and hence the single chain approximation is quite reasonable. However, infra-red studies on PVA 1° show that strong intermolecular H-bonding occurs between adjacent chains, but no intramolecular H-bonding (in the chain direction) occurs. Sakurada 9 also noted that H-bonding affected only the modulus perpendicular to the chain. As the PVA chain is extended it tends to contract laterally and when two adjacent chains are extended this will result in the H-bonds between them being stretched. Therefore the H-bond force constant also contributes to the elastic modulus in the chain direction and the single chain approximation becomes invalid. Jaswon et al. 1 gave a treatment for secondary bonding in cellulose and it is hoped to extend this to PVA in the near future. The elastic modulus of s-polyacrylonitrile was calculated as 236 G N / m L No experimental or calculated moduli were available for comparison, but the f value obtained was well outside the expected range for a planar zig-zag polymer (see Table 4). The force constants used were obtained from a recent detailed treatment of the alkylnitriles n and are regarded as reliable. The high extensibility value is due to the extremely polar nature of the polymer. Polyoxymethylene (POM) also exhibits strong intramolecular dipole-dipole interactions resulting in high modulus and extensibility values 9. The calculated values for the modulus a n d f v a l u e of s-acrylonitrile, therefore appear to be quite reasonable. The low modulus and f values obtained for s-PMMA are shown in Tables 3 and 4. The f value is only slightly lower than the expected range and is due to the uncertainty in the calculation for the cross-sectional area of the chain using X-ray density measurements rather than unit cell parameters.
ACKNOWLEDGEMENT We thank the Science Research Council for a scholarship to C.G.M. REFERENCES 1 Jaswon, M. A., Gillis, P. P. and Mark, R. E. Proc. R. Soc. (A) 1968, 306, 389 2 Manley, T. R. and Martin, C. G. Polymer 1971, 12, 775 3 Shimanouchi, T., Asahina, M. and Enomoto, S. J. Polym. Sci. 1962, 59, 93 4 Treloar, L. R. G. Polymer 1960, 1, 95 5 Miyazawa, T. Rep. Progr. Polym. Phys. Japan 1965, 8, 47 6 Asahina, M. and Enomoto, S. J. Polym. Sci. 1962, 59, 101 70verend, J. and Scherer, J. R. J. Chem. Phys. 1960, 32, 1289, 1296, 1720 8 Shimanouchi, T. J. Chem. Phys. 1949, 17, 245, 734, 848 9 Sakurada, I. and Kaji, K. J. Polym. Sci. (C) 1970, 31, 57 10 Kuhn, L. P. J. Am. Chem. Soc. 1954, 76, 4323 11 Fujiyama, T. Bull. Chem. Soc. Japan 1971, 44, (1), 89 12 Bunn, C. W. Trans. Faraday Soc. 1939, 35, 482
APPENDIX Derivation of the force constants K and H Assuming a Urey-Bradley type field for polyethylene, the terms in the potential function involving the coordinates of the nth CH2 group are given asr: ' Arn + Ar~) + ½KeE[(Arn)2 + (Ar')2] + Vn = Kr~a( KScb( ARn) + ½Kcc( ARn) 2 + H~cFia2( AOn) + ½H~cna2( AOn) 2 + H~ccb2( Agan) + ½Hcccb2( Agan)2 + HHccab(A~n + A¢~ + A C n. .+. ±. ¢ . )+ ' ½Hnccab[(A¢,)2 + (A ¢~)~ + (A¢~)z + (A¢~,") ~] + F{i~(2asin~)±qn + ½FHn(Aqn) 2 + F~c( 2bsin~)A Qn + ½Fcc( A Qn) 9"+
F & 4 A p . + AP;, + :xPT,+ AP~,"] +
½FcH[(AP.)2 + (Ap~,)~+ (Ap;;)~+ (Ap;")2]
(A1)
where internal coordinates R~, r~ etc., denote the distances and angles respectively with equilibrium values as indicated in parenthesis: R n ( C n - Cn+l =b) r n ( C n - H n = a ) r'~(Cn-H'.=a) Cn( < C n + l - C n - Cn-1 = 2fi)
On( < Hn - Cn - H~ = 2=) Cn(< H n - C n - Cn+l =2~) ¢;,( < H~, - Cn - Cn+l = 2y) ~b~( < n n - C n - C n - l =
2y)
¢~,"( < H~, - Cn - Cn-1 =
2~)
Qn(Cn_l . . . Cn+l = 2bsinfi)
qn(Hn • • • H~,=2asin~) Pn(Hn • • • Cn+l = ~) P,~(H~,... Cn+l= ~) Pn(Hn...
Cn-1 = ~)
P ~ " ( H ~ . . . C n - l = a) where a2 = a 2 + b 2 _ 2abcos2y and cos27 = - cosacosfl t/
When the coordinates Qn, qn, P ' , Pn and P~" for changes of distances between non-bonded atoms are expressed
POLYMER, 1973, Vol 14, O c t o b e r
495
Elastic modulus of linear polymer crystals: L R. Manley and C. G. Martin in terms of the other coordinates 6 the potential energy becomes:
K n = - t ~Fh rr + s~Faa K22 = - t~F~c + s~Fcc
2 Vn = K1 [(Arn) 2 + (Ar~) 2] + K2(ARn) 2 +
K12 = -- tst4F~c + SaSaFHc
Fn = tlSl(F~a + Far O F22 = t2sz(F'cc + Fcc) F13 = (b/a)i/2(t3s4F~c + s3t4Fac) F23 = (a/b) al2(t4saF~c + s4tsFr~c) Hla=K/a(2ab) l/2 H2a = K/b(2ab) 1/8
Hl(aAOn)2 + I-I2(bA~n)2 +
Haab[(A~bn)2+ (A~.) , 2 +(A~.) " 2 +(A~.'" ) 2]+ 2Kn( Arn)( Ar~) + 2K22(ARn)( AR.-1) + 2K12[( Arn)( A Rn) + ( Ar~)( ARn) + ( Arn)( ARn-1) + (Arn)(ARn-x)] +
2Fn(aaO.)( Ar. + Ar~) + 2F22(bA$.)( AR. +
n s a = K/2112ab
AR.-1) + and
2Fla(ab) Z/2[(Arn)( A~bn) + (Ar,~)(A~b~)+
( Ar.)(A 4,;;)+ (Ar~)(A¢~'3] + 2F2a(ab)~/2[(AR.)(A¢.) + (AR.)(A ¢~) + (AR._I)(A ~:) + (AR._z)(A~b~")]+
sl = sina, sz = sinfl
ss=(a-bcos2y)/a s4 = ( b - acos2y)/a
2H13a(ab) 1/2[(AOn)(A~bn) + (A 0n)(A~b~,)+
t l = COSo~
(±o.)(A~.)tt +(A 0 .)(a~.t i t )] +
t2 = c o s t ta = bsin2~,/a
2H2ab(ab) I/2[( A~n)( A~bn) + (A~)(A~b~) +
(A~.)(A ~ ) + (A¢.)(A¢' 3] + 2I-Iaaab[(A~bn)(A~b;) + (A~bn)(A~b~)+
(A~)(~")+(A~))(A~;,")] where 2 ' Kl = Kcr~ + t~Frla + S~~ F rlI-i+ 2t~F~c + 2s~Fac /(2 = Kcc + 2t~F~c + 2s~Fcc + 4t~F~c + 4s~Fac HI = HI~ c r i - s~F~iri + t ~Far~ + (3x/81/2a2) //2 = Hccc - s ~ F ~ c + t2Fcc + (3~/81/2b 2) Ha = H a c c - sasaFhc + tataFac + (3,~/8 ~/2ab)
496 POLYMER,1973, Vol 14, October
(A2)
t4=asin2~,/a Since in our modulus calculation we are considering the small changes in C n - C n - z , i.e., Rn and in C n + I C n - f n - 1 , i.e., ~n the required force constants are K2 and//2. •
K=/£2 = Kcc + 2s~Fcc + 4s]Fnc
(A3)
and //2 = Hccc + t~Fcc + 3K/81/2b2
(A4)
neglecting the first order terms since they have little effect on the value of the modulus.
Treloar who used a statical analysis over the whole molecule. Both methods are much simpler than those of Shimanouchi and Miyazawa, the latter being based on the complete potential function for the polymer. Using force constants based on a Urey Bradley force field, the following values of modulus were found (all syndiotactic except PE) in GN/m2: PE, 299; PVC, 153; PVF, 212; PVA, 142; PAN, 236; PMMA, 63.
INTRODUCTION The method of Jaswon et al. 1 for the calculation of the elastic moduli of cellulose was modified and applied to the helical polymer crystals of poly(phosphonitrilic chloride) 2 and values for Young's modulus of 0"138 GN/m e and 1.66 GN/m ~ obtained for the uniform helix and cis-planar structures respectively. These results are similar to those calculated for other helical polymers, but as all experiments with (NPCI2)n used amorphous rubber, no direct comparison is possible. The method is equally applicable to linear polymers and it was therefore decided to apply it to crystals of polyethylene and the syndiotactic forms of poly(vinyl chloride) (PVC), poly(vinyl fluoride) (PVF), poly(vinyl alcohol) (PVA), polyacrylonitrile (PAN) and poly(methyl methacrylate) (PMMA). Different solutions have been proposed for both polyethylene3-5 and PVC 6. Treloar 4 performed a statical analysis using simple valence force constants. On the other hand, the Japanese workers 3, 5,6 calculated the modulus directly from the complete potential function for the molecule. We assume that the polymer chain in the crystal is isolated and infinite; the repeat unit alone is characterized. The problem of a free or fixed end group does not arise. The bond lengths and bond angles of the repeating unit are taken as degrees of freedom which can be varied independently of one another. This simplifies the choice of force constants since we derive the extensions due to bond stretching separately from those due to angular deformation. The force constants are selected from a Urey Bradley force field (UBFF), ih which only the 'pure' stretching or bending constants for the appropriate bond are taken. The mixed terms, i.e. bending-stretching interaction constants are meaningless in our method of modulus calculation. In obtaining these force constants we do, however, consider the effect of adjacent nonbonded atoms on the particular stretching or bending force constant in question. The number of force constants required for our analysis is a minimum and they a r e calculated from a knowledge of K, H and F, the stretching, bending and repulsive force constants for the
appropriate bonds or angles, which may be transferred from smaller molecules 7. The methods of Shimanouchi et al. 3, 6 and Miyazawa5 require the setting up of the complete Urey Bradley force field for the polymer repeat unit including both stretchingstretching and stretching-bending interaction terms. The relationships between internal parameters and the helical parameters of the particular polymer structure are also required. Some knowledge and expertise in spectroscopic techniques are required in order to set up the potential energy equation, remove the redundancies from it and derive the relationships between the helical and internal parameters. The calculation of the modulus requires the differentiation of all these equations and the solution of the resultant simultaneous equations. The present method, therefore, is much simpler. This advantage is particularly important with polymers more complex than polyethylene. The Japanese workers 5, ~ have treated polymers with up to four different chain units, e.g. -(-A1--A2--A3--A4-)-n but excluding -(-A1--A2--[Aa]x-)-n. On the other hand, the present method is readily applicable to polymers such as:
-(-A1--A2--[Aa]x--A z--A1-- [A3]y~)-n POLYETHYLENE The polyethylene molecule has an extended planar zig-zag structure in which the repeating unit is the --CH~-- group. This is characterized by the bond length r and bond angle ¢ (see Figure 1). The translational repeat unit is defined as SS' and is made up from ½rl, ~, r2, ¢, ½rl, (Figure 2). It is more convenient, however, to base it on C1C'1 bearing in mind that ¢ occurs twice. Both definitions give the same equations but the latter is more convenient. A set of rectangular Cartesian axes related to these quantities can be superimposed by choosing one of the carbon atoms as an origin and identifying one of the bonds as the direction of the x axis as shown in Figure 2. The y axis is then perpendicular to C1C2 and is in the
POLYMER, 1973, Vol 14, October 491
Elastic modulus of linear polymer crystals: T. R. Manley and C. G. Martin I
l
Hn-2 Hn-2
Now from equation (1)
l
Hn Hn
Hn+2.Hn÷2
•
-dO
i.e.
dLa = ~/2[sin0, - cos0]d¢ dLa = (cosf~sin0- sint2 cos 0)12d¢
•
(7)
Then utilizing the Jaswon et al. formulai:
Hn-i Hn-i
Hn÷lHn+l
C i ~ / O L \ 2 1 02W - =~) /O(dl,)2
Figure 1 Structure of polyethylene molecule
Y
the overall force constant C for the repeat unit can be obtained from: C-i=
S
(dLi) 2 , (dL2) 2 , (dLs) 2 K -t'~-t 2H
(9)
where K and H are the bond stretching and bond angle deformation force constants for - C H z - C H s - and -CH2-CH2-CH2- respectively. The factor 2H arises since the degree of freedom ff occurs twice in the repeat distance, (cf. SS' in Figure 2). The overall force constant C is then related to the elastic constant csz by the energy balance:
-"-X
C~
(8)
C2
/ Figure 2 Translational repeat unit of PE
½cs2e~V= W= ½C(dL) 2= CLSe~
(lO)
plane of the molecule. The z axis is out of the paper but is not required for this linear planar molecule. The auxiliary angle 0 is the angle between bond C2C~ and the x axis and is related to ff by" ¢=,~-0 (1) Vector components can now be assigned to the bonds in the repeat unit: CiC2 = (1, 0)ri
where A is the effective cross-sectional area of the repeating unit. The force constants used were taken from a UBFF, the deviation of which is shown in the Appendix and are in the form:
C2C ~= (cos0, sin0)r2
K = 1£2= Kcc + 2s~Fcc + 4s~Frtc
•
CL2 CL cs2= V = A
(11)
and
The repeat trait vector is given as: C1C~= g. L = (cosf2, sinf~)L
L = (r~ + r ~ - 2fir s cos~) i/9'
(3)
When the chain is deformed by a uniform strain along its axis the symmetry is still preserved and hence the three quantities ri, rs and ¢, still describe the chain configuration. These can be considered as three degrees of freedom which can be varied independently of one another and will be designated as h using our previous notation s. If we consider the repeating unit L to undergo a change in length dL, this change is made up of the sum of the small changes due to a small change in each degree of freedom, i.e. dL =~dL~; i = 1, 3 (4) These small changes are given as follows: dLi = ~(1, 0)dli = cost2dli
H=b2Hz
(2)
where f~ is the angle between the fibre direction along CiC'i and the x axis and L is the repeat distance given by:
where
1-I2= Hccc + t~Fcc + 3K/(81/SbS)
= (cosf~,cos0 + sinf~, sin0)dls
SYNDIOTACTIC PVC Syndiotactic PVC has an extended planar zig-zag structure as shown in Figure 3 and the modulus may be calculated in the manner used for PE.
(5) (6)
and dLa = ~/2:¢[cosO, sin0]d4, dO
(I) CH2 Figure 3 Structure of s-PVC
492 POLYMER, 1973, Vol 14, October
(13)
Force constant data were taken from refs 3 and 8, since no complete set of UBFF constants was available for the polyethylene molecule. The molecular parameters are given in Table 1 and the values used in the force constant calculations in Table 2. The modulus (c22) for polyethylene is calculated as: 299 GN/m 2.
dLz = a(cos0, sin0)dl2
dO
(12)
qaa
(I) CH2
Elastic modulus of linear polymer crystals: T. R. Manley and C. G. Martin Table 1 Molceular parameters used in modulus calculations Tetrahedral bond angles are used throughout (109° 28') C-C bond distances are all 0.154 nm
Polymer (CH=CXY)n
X
Y
R(C-X) (nm)
R(C-Y) (nm)
Crosssectional area,* A (nm 2)
PEa s-PVCb s-PVFc s-PVAc s-PANc s-PMMAc
H CI F OH CN COOCH3
H H H H H CHa
0.109 0.177 0.135 0"413 0"216 0"154
0'109 0.109 0.109 0-109 0-109 0.154
0.1824 0"2746 0.2121 0-2152 0-3111 0'5565
Bond distances
Nature of
Repeat distance L" (nm) 0.2515 0-2515 0'2520 0.2520 0-2550 0.2510
* From unit cell measurements except for PMMA a Ref. 12 b Ref. 6 c Alexander, L. E. 'X-ray diffractions in polymer science'. Wiley, N.Y. 1969, p. 473 Table 2 Force constant data used in modulus calculations Nature of Polymer (CH¢CXY)n PEa,b s-PVCc s-PVFc s-PVAd s-PANe s-PMMAf
X
KC-C (N/m)
Y
FCH2-CHz FCXY-CXY (N/m) (N/m)
FCX (N/m)
FCy (N/m)
Hi
H2
(N/m)
(N/m)
(aNm) 0'2 0"23* 0-25* 0'17 Kz=0"01 ~=0"11 0'1
H
H
CI F OH CN
H H H H
280 340 340 280 200
96 33 33 33 38
96 96 150" 30 44
40 60 130 28d 44
40 40 40 40 54
11 11 11 20* 34
11 20 25* 11 27
COOCH3
CHz
340
96
33
83
40
11
20
* Estimated value a Ref. 3 b Ref. 8 c Ref. 6 d From propyl alcohol e Ref. 11 f From polyisobutylene and polypropyleneS
or in this case ,
OS ~, CH(2) J r 2 1
L
L
00 = (cosfl, sinfl) 2 = ~ 2 = ~L
) .'.
.~Ir12 t-u
,
dL1 = costld6
(14)
dL2 = ~(cos01, sin01)dl2 = (cost) cos01 + singl sin01)d/2
O
m
X
/ Figure 4 Translational repeat unit for s-PVC
The rectangular Cartesian axes are superimposed as for PE with a C atom as origin and a bond r21 as the x axis as shown in Figure 4. Although there are four C atoms in the repeat unit we are only considering deformation of the carbon skeleton and since ¢1=¢2 the alternate nature of the chlorine atoms along the chain will be mechanically indeterminate• Hence we need only consider half of the actual repeat unit with four degrees of freedom to be varied rzl, rlz, ¢1 and ¢2 and defining the auxiliary angles 01 and 02. The mechanical model is, therefore, almost exactly the same as for polyethylene and gives the following expressions for the small changes dL~.
(15)
which are exactly the same as in polyethylene. The angles ¢1, i.e,
,"
d01 = - d¢1 r12~
dLa= 24,~w
~-~
sin01-sint) cos01)d¢l
(16)
and also d_ _r12 0 La=n 2 0¢~ [-c°s0z'-sin0z]dCz -ra2rt - sin0z, cos0z]dCz dL4 = n~-
dLi = ~(1, O)dh where ~ is the direction cosine of the repeat distance vector Off' = (cosQ, sinfl)L = ri. L
since
d¢2 = - d02
.'.
dL4= r1,,2[- cost) sin0,~ +sinfl cos02]dCz z
POLYMER, 1973, Vol 14, October
(17)
493
Elastic modulus o[ /inear polymer crystals: T, R. Manley and C. G. Martin The overall force constant C for the repeat unit can then be obtained from:
Table 3
Calculated moduli of several polymer crystals
Calculated results
C-1
(GN/m~)
(dL1) 2 , (dL2) 2 . (dLs)2~_(dL4) 2 =
K
-
(18)
where K and H~x, H~ are the stretching and bending force constants for -CHCI-CHz--, - C H C I - C H 2 - C H C I - and - C H z - C H C I - C H 2 - respectively. DERIVATION OF THE FORCE CONSTANTS K, H~, H~ The Urey Bradley force field (UBFF) used for PVC is similar to that for polyethylene given in the Appendix but contains more terms due to the difference between CH2 and CHC1, i.e. the term in ARn in polyethylene ( C n - C n + l ) becomes a term in Ar12 and Ar21. Similarly the term in Affn becomes a term in A9~1and A¢2. Thus the required force constants are: K = K rl, = K r- = Kcc + S~fCa,CH, + s~2FcHelCHCt +
s~2Fccl+ 3s~zFcrI (19) and
H~ = p2H1 + t22p2FcHc1CHC1+ p 2 3 x / ( 8 l12r21r12)
(20)
H~=p2H2+t~1p2FcH,CH=+p23K/(81/2r12r21)
(21)
p
q
PE
999 189
s-PVC s-PVF s-PVA
153 219 142 236 63
340a 182b 290c 160d
s-PAN s-PMMA
Experimental results 9 (GN/m 2) 235
251
a Ref. 3 b Ref. 4 c Ref. 5 dRef. 6 pThis work q Other authors
where Z is the number of monomer units per unit cell (in this case Z = 4), M = molecular weight of a monomer unit (100.11), V= volume of the unit cell, N = Avogadro's n u m b e r = 6.023 x 1023 mo1-1, and the chain length L = 5.02014 x 10 -8 cm. The only X-ray density available for s-PMMA was 1.19 g/cm a obtained from the amorphous region. The crystalline density will be slightly higher but for the approximate nature of this calculation for cross-sectional area the value was thought to be adequate. The crosssectional area of the 'unit cell' was thus calculated as 111.291 × 10-16 cm 2 giving an effective cross-sectional area of chain as 55"645 x 10-16 cm 2. Substituting the above in equations (14) to (22) gave the modulus of s-PMMA as c22 = 63 G N / m L This low value of the modulus is attributed to the large cross-sectional area of the c h a i n = 5 5 x 10-16 cm 2 compared with 18.24x 10 -16 cm 2 for polyethylene, 27.46 x 10-16 cm 2 for PVC, etc., which is caused by the bulky methyl and ester groups in the side chain.
where sn = (r12 - r21cos¢2)/qn
s 2z = (r21- r12cos¢l ) / q22 s12 = (r12- r~cos¢')/q12 S21 = (r21 - -
Polymer
r~cos¢~)/q~t
tn = psin$2/qn t22 = psinfl/q22 and
p=(r12r21) 1/2 2_ 2 2-2r12r2tcos$2 qll--rl2+r21 q 22.9= r ~1+ r ~2- 2r21rl2cos$1 12 2 t2 ~ t .t q 12 = r 12 + r 2 - - z r 1 2 r 2 c ° s 9 2 '2
2
'2
q21= r 21+ rl
EXTENSIBILITY
-- 2r21r l' C O S ~ I
Using the values in Tables I and 2 the elastic constant c22 of s-PVC was calculated as c22 = 153 G N / m z. The simplicity of this method for calculating the elastic moduli of planar zig-zag polymer chains of type (A1-A2-A1-A2) enabled us to calculate the moduli of other linear polymers, e.g. PVF, PVA and PAN, by substituting data into equations (14) to (21) as for the s-PVC calculation. The parameters from Tables 1 and 2 gave the results shown in Table 3.
The values of modulus for polymers of type (A1-A2)n in Table 3 were used to calculate the extensibility o r f v a l u e . The extensibility is the force required to stretch a molecule by 1 ~o in the direction of the molecular axis and may be calculated from the modulus and cross-sectional area of a single chain 9. The f value is mainly dependent on conformation and for planar zig-zag type polymers usually falls in the range 0.4--0.5 nN. The f values for each of the polymers are shown in Table 4. DISCUSSION
SYNDIOTACTIC
POLY(METHYL
METHACRYLATE)
In this case very little X-ray information about the unit cell is available so the cross-sectional area of the unit cell was calculated from the X-ray density-volume relation:
ZxM
density= Vx N
494 POLYMER, 1973, Vol 14, October
(22)
The value for polyethylene obtained by the present method is identical to that of Treloar 4 if the same force constants are employed (Table 3). The improved values for the force constants in the present paper give a result lower than that of Shimanouchi 3 but close to that of Miyazawa 5. A much better result is expected when force constant values from a single source are available. As
Elastic modulus of linear polymer crystals: T. R. Manley and C. G. Martin Table 4
Extensibilities (fvalues) of several polymer crystals Modulus
Cross-sectional area
Polymer
(GN/m 2)
(nm 2)
/value (nN)
PE s-PVC s-PVF s-PVA s-PAN s-PMMA
299 153 212 143 236 63
0" 1824 0"2746 0.2121 0.2152 0.3111 0-556
0' 545 0-420 0.449 0.31 0"73 0"35
expected, the theoretical value of the elastic modulus in each case is higher than the experimental value obtained using X-ray methods 9. The present method is preferred because it is simplest. For s-PVC, this method gives a value close to that calculated by Asahina and Enomoto 6 (see Table 3). No X-ray data were available for PVC, but the f value obtained is within the expected range (see Table 4). Similarly, no experimental or calculated data are available for s-PVF but the extensibility falls within the expected range (see Table 4). Syndiotactic PVA gave a low result when compared to the experimental result (see Table 3). This could be due to lack of suitable force constants for substitution into the UBFF, but is more likely to be due to neglect of secondary bonding effects. We consider an isolated polymer chain, i.e. neglecting intermolecular forces. These are normally very small when compared with the primary valence forces along the chain and hence the single chain approximation is quite reasonable. However, infra-red studies on PVA 1° show that strong intermolecular H-bonding occurs between adjacent chains, but no intramolecular H-bonding (in the chain direction) occurs. Sakurada 9 also noted that H-bonding affected only the modulus perpendicular to the chain. As the PVA chain is extended it tends to contract laterally and when two adjacent chains are extended this will result in the H-bonds between them being stretched. Therefore the H-bond force constant also contributes to the elastic modulus in the chain direction and the single chain approximation becomes invalid. Jaswon et al. 1 gave a treatment for secondary bonding in cellulose and it is hoped to extend this to PVA in the near future. The elastic modulus of s-polyacrylonitrile was calculated as 236 G N / m L No experimental or calculated moduli were available for comparison, but the f value obtained was well outside the expected range for a planar zig-zag polymer (see Table 4). The force constants used were obtained from a recent detailed treatment of the alkylnitriles n and are regarded as reliable. The high extensibility value is due to the extremely polar nature of the polymer. Polyoxymethylene (POM) also exhibits strong intramolecular dipole-dipole interactions resulting in high modulus and extensibility values 9. The calculated values for the modulus a n d f v a l u e of s-acrylonitrile, therefore appear to be quite reasonable. The low modulus and f values obtained for s-PMMA are shown in Tables 3 and 4. The f value is only slightly lower than the expected range and is due to the uncertainty in the calculation for the cross-sectional area of the chain using X-ray density measurements rather than unit cell parameters.
ACKNOWLEDGEMENT We thank the Science Research Council for a scholarship to C.G.M. REFERENCES 1 Jaswon, M. A., Gillis, P. P. and Mark, R. E. Proc. R. Soc. (A) 1968, 306, 389 2 Manley, T. R. and Martin, C. G. Polymer 1971, 12, 775 3 Shimanouchi, T., Asahina, M. and Enomoto, S. J. Polym. Sci. 1962, 59, 93 4 Treloar, L. R. G. Polymer 1960, 1, 95 5 Miyazawa, T. Rep. Progr. Polym. Phys. Japan 1965, 8, 47 6 Asahina, M. and Enomoto, S. J. Polym. Sci. 1962, 59, 101 70verend, J. and Scherer, J. R. J. Chem. Phys. 1960, 32, 1289, 1296, 1720 8 Shimanouchi, T. J. Chem. Phys. 1949, 17, 245, 734, 848 9 Sakurada, I. and Kaji, K. J. Polym. Sci. (C) 1970, 31, 57 10 Kuhn, L. P. J. Am. Chem. Soc. 1954, 76, 4323 11 Fujiyama, T. Bull. Chem. Soc. Japan 1971, 44, (1), 89 12 Bunn, C. W. Trans. Faraday Soc. 1939, 35, 482
APPENDIX Derivation of the force constants K and H Assuming a Urey-Bradley type field for polyethylene, the terms in the potential function involving the coordinates of the nth CH2 group are given asr: ' Arn + Ar~) + ½KeE[(Arn)2 + (Ar')2] + Vn = Kr~a( KScb( ARn) + ½Kcc( ARn) 2 + H~cFia2( AOn) + ½H~cna2( AOn) 2 + H~ccb2( Agan) + ½Hcccb2( Agan)2 + HHccab(A~n + A¢~ + A C n. .+. ±. ¢ . )+ ' ½Hnccab[(A¢,)2 + (A ¢~)~ + (A¢~)z + (A¢~,") ~] + F{i~(2asin~)±qn + ½FHn(Aqn) 2 + F~c( 2bsin~)A Qn + ½Fcc( A Qn) 9"+
F & 4 A p . + AP;, + :xPT,+ AP~,"] +
½FcH[(AP.)2 + (Ap~,)~+ (Ap;;)~+ (Ap;")2]
(A1)
where internal coordinates R~, r~ etc., denote the distances and angles respectively with equilibrium values as indicated in parenthesis: R n ( C n - Cn+l =b) r n ( C n - H n = a ) r'~(Cn-H'.=a) Cn( < C n + l - C n - Cn-1 = 2fi)
On( < Hn - Cn - H~ = 2=) Cn(< H n - C n - Cn+l =2~) ¢;,( < H~, - Cn - Cn+l = 2y) ~b~( < n n - C n - C n - l =
2y)
¢~,"( < H~, - Cn - Cn-1 =
2~)
Qn(Cn_l . . . Cn+l = 2bsinfi)
qn(Hn • • • H~,=2asin~) Pn(Hn • • • Cn+l = ~) P,~(H~,... Cn+l= ~) Pn(Hn...
Cn-1 = ~)
P ~ " ( H ~ . . . C n - l = a) where a2 = a 2 + b 2 _ 2abcos2y and cos27 = - cosacosfl t/
When the coordinates Qn, qn, P ' , Pn and P~" for changes of distances between non-bonded atoms are expressed
POLYMER, 1973, Vol 14, O c t o b e r
495
Elastic modulus of linear polymer crystals: L R. Manley and C. G. Martin in terms of the other coordinates 6 the potential energy becomes:
K n = - t ~Fh rr + s~Faa K22 = - t~F~c + s~Fcc
2 Vn = K1 [(Arn) 2 + (Ar~) 2] + K2(ARn) 2 +
K12 = -- tst4F~c + SaSaFHc
Fn = tlSl(F~a + Far O F22 = t2sz(F'cc + Fcc) F13 = (b/a)i/2(t3s4F~c + s3t4Fac) F23 = (a/b) al2(t4saF~c + s4tsFr~c) Hla=K/a(2ab) l/2 H2a = K/b(2ab) 1/8
Hl(aAOn)2 + I-I2(bA~n)2 +
Haab[(A~bn)2+ (A~.) , 2 +(A~.) " 2 +(A~.'" ) 2]+ 2Kn( Arn)( Ar~) + 2K22(ARn)( AR.-1) + 2K12[( Arn)( A Rn) + ( Ar~)( ARn) + ( Arn)( ARn-1) + (Arn)(ARn-x)] +
2Fn(aaO.)( Ar. + Ar~) + 2F22(bA$.)( AR. +
n s a = K/2112ab
AR.-1) + and
2Fla(ab) Z/2[(Arn)( A~bn) + (Ar,~)(A~b~)+
( Ar.)(A 4,;;)+ (Ar~)(A¢~'3] + 2F2a(ab)~/2[(AR.)(A¢.) + (AR.)(A ¢~) + (AR._I)(A ~:) + (AR._z)(A~b~")]+
sl = sina, sz = sinfl
ss=(a-bcos2y)/a s4 = ( b - acos2y)/a
2H13a(ab) 1/2[(AOn)(A~bn) + (A 0n)(A~b~,)+
t l = COSo~
(±o.)(A~.)tt +(A 0 .)(a~.t i t )] +
t2 = c o s t ta = bsin2~,/a
2H2ab(ab) I/2[( A~n)( A~bn) + (A~)(A~b~) +
(A~.)(A ~ ) + (A¢.)(A¢' 3] + 2I-Iaaab[(A~bn)(A~b;) + (A~bn)(A~b~)+
(A~)(~")+(A~))(A~;,")] where 2 ' Kl = Kcr~ + t~Frla + S~~ F rlI-i+ 2t~F~c + 2s~Fac /(2 = Kcc + 2t~F~c + 2s~Fcc + 4t~F~c + 4s~Fac HI = HI~ c r i - s~F~iri + t ~Far~ + (3x/81/2a2) //2 = Hccc - s ~ F ~ c + t2Fcc + (3~/81/2b 2) Ha = H a c c - sasaFhc + tataFac + (3,~/8 ~/2ab)
496 POLYMER,1973, Vol 14, October
(A2)
t4=asin2~,/a Since in our modulus calculation we are considering the small changes in C n - C n - z , i.e., Rn and in C n + I C n - f n - 1 , i.e., ~n the required force constants are K2 and//2. •
K=/£2 = Kcc + 2s~Fcc + 4s]Fnc
(A3)
and //2 = Hccc + t~Fcc + 3K/81/2b2
(A4)
neglecting the first order terms since they have little effect on the value of the modulus.