The Melting Points of Oligomethylenes

TON THAT MlNH TAN and BERND MICHAEL RODE*

Institute for General, Inorganic and Theoretical Chemistry, University of Innsbruck, lnnrain 52A, A-6020 Innsbruck, Austria

SYNOPSIS

Quantitative structure-property relationships (QSPR) between the melting points of oli- gomethylenes and quantum chemical data calculated by the semiempirical method CNDO/ 2 were investigated. A linear relationship between the average net charges of hydrogen and carbon atoms in the molecules and the melting points could he established, achieving good fit as well as predictability. Further, the relation between the net charges and the chain length was evaluated. The evaluated net charges of polyethylene lead to a calculated melting point of 141.4OC (experimental: 141.4'C). 0 1996 John Wiley & Sons, Inc. Keywords: quantum chemistry CNDO/2 oligomethylene polyethylene melting point

I NTRO DUCT10 N

The physical properties of oligomethylenes, es- pecially their melting points, are of much practical interest for polymer chemists. Therefore, the pre- diction of melting points (T,) has been studied by many authors. The most popular equations for this prediction are based on the assumption that the melting points of oligomethylenes are dependent on the contribution of end groups presented by an ad- ditive term taken to be independent of the chain length n

T, = AH*(n + a)/AS*(n + b) (1)

where a, b are constants, and AH* and AS* represent the enthalpy and entropy of melting per CH, group in the limit n = 03, under the assumption that AH* and AS* are not dependent on the chain length. Re- placing AH*/AS* by To, eq. (1) transforms to

T, = To(n + a ) / ( n + b)

where To is defined as

To = lim T,(n) ( 3 ) n-r cc

* To whom correspondence should be addressed. Journal of Polymer Science: Part B: Polymer Physics, Vol. 34,2139-2143 (1996) 8 1996 John Wiley & Sons, Inc. CCC OS87-6266/96/132139-05

The values of a, b, and To were set to a = -1.5, b = 5.0, and To = 141.1 C by fitting experimental data of ~ligomethylenes.~ To represents the melting point of polyethylene (PE). Later Broadhurst4s5 modified eq. (1) by introducing an additional term R In n de- fined by Flory and Vrij' and thus adapted eq. (1) to the following form

T, = To(n + a ) / ( n + R In n + b) (4)

To in eq. (4) as obtained by fitting experimental melting points of oligomethylenes is 144.8 C, whereas it should actually represent the melting point of polyethylene for the case n + co. Since it is significantly larger than the experimental value7 of 141.4 C, Somayajulu' modified Kreglweski's equations,' which are based on the theory of Flory et al.," to determine the melting point of oligometh- ylenes with n > 30. Kreglewski's equations for boil- ing point and critical temperature are

ln[(T," - Tb)] = a - bn2l3

ln[(TP - T,)] = a - bn2/3

(5)

(6)

where a, b are constants, and Tbr TF, T,, and TF are boiling point and critical temperature of oligo- methylenes and PE, respectively. Somayajulu's equation for the melting point is

2139

2140 TAN AND RODE

ln[(Tg - T,)] = a - bn1’25 (7)

where T,” is the melting point of PE and T,,, is that of an oligomethylene. The melting point predicted for PE by this approach is 141.4”C.

The melting point is affected by the degree of flexibility of the molecular chain and the cohesive forces. In oligomethylene where the linear molecular chain consist of simple basis units containing only carbon and hydrogen atoms, the cohesive forces be- come the most important factor affecting the melting points. The dipole moments caused by the atomic charges also contribute-besides dispersion forces- to this cohesive energy, thus suggesting a correlation between the charges and the melting points as well. Therefore, a linear relationship between atomic charges available from quantum chemical calcula- tions by a semiempirical method (CNDO/2) and the melting points of oligomethylenes was investigated. The relation between the net charges calculated by the CNDO method and the chain length was also investigated by statistical methods to evaluate the net charges for PE. Finally, the melting point of PE was calculated.

METHOD

The melting points of 63 oligomethylenes ( n = 6- 6000) and polyethylene were collected from the lit-

The molecular geometries were en- ergy optimized using the force-field program AL- CHEMY Ill l4 with the assumption that they are in an all-trans conformation. The electronic charges of atoms in the optimized molecules were then eval- uated by Mulliken population analysis following MO-SCF calculations of CNDO16 type (original parametrization) by the CNDO/KGN program.17 The absolute value of such charges strongly depends on the method used, even in case of ab initio cal- culations, where they are basis-set dependent. Their relative values in a series, however, should reflect structure-dependent charges in a very similar way for either ab initio or semiempirical methods. As ab initio calculations are extremely computer intensive for large molecules, semiempirical methods seemed to be the most appropriate means, and among them the “classical” CNDO method, parent of all newer methods (MNDO, AM1, PM3) , parametrized for various specific purposes, was chosen.

The compounds were divided into two groups. The first group consists of oligomethylenes contain- ing an even number of carbon atoms and was used for establishing the QSPR model for melting points.

The average net charges of carbon atoms (6) and hydrogen atoms ( Z ) in the molecule were calculated as

n

6‘ = 2 Ci/n i

where Ci , Hi are the net charges of the i th carbon atom and ith hydrogen atom in a molecule, respec- tively, and n is the number of carbon atoms in a molecule.

These average charges 6 and 3 were then used to establish a linear model for predicting the melting point according to the formula:

where bo is a constant and bl, b2 are the model pa- rameters.

The quality of the model was evaluated by cor- relation coefficient ( R 2 ) , standard error (S) and significance based on the F - t e ~ t . ~ ~ , ’ ~

The second group, consisting of the molecules having an odd number of carbon atoms, was used to evaluate the predictability of the model.

The S / r ratio and RVvalue determine fitting and prediction errors:

S / r = standard error ( S ) / range of observation ( r ) ( 11)

RV = 2 (observed-predicted value) ’/

(number of compounds - 1 ) . ( 12 )

A high-quality model should give low values for both S / r ratio and RV.

In a last step, the net charges of PE were calcu- lated by statistical analysis and the melting point of PE was evaluated on the basis of the previously established model.

RESULTS AND DISCUSSION

The model for the melting point of oligomethylenes established from the first group leads to the following equation

T,,, = 484.4 + 278687.86‘ + 503772.43. (13)

From the equation i t is obvious that the melting point of oligomethylenes is influenced by the charge of hydrogen atoms to a different extent than by that of carbon atoms. However, for very long chain oli- gomethylenes @ must be - 2 2 , i.e., both variables are fully correlated and only one of them should be sufficient for the model. Models using either only @ or only 2 were also evaluated.

I t can be seen from Table I that R 2 for the model using both charges is closed to 1 (0.99835) and its standard error S is very small ( 1.72282), indicating that the model has a very good fit. The F-value of the model is very high (81498.45) and, therefore, the model is highly significant a t 0.01 level. Table I also shows the statistical characteristics of the mod- els using only @ or only 2. The slight advantage of the model using both charges can be ascribed to the fact that the additional H atoms a t the terminal group still exert some influence in short-chain oligomers, leading to an inequality of @ and - 2 2 , respectively. For this reason, it seems convenient to use both @ and 2 in the model in order to describe short- and long-chain oligomethylenes with equal quality.

The quality of the fitting process for the com- pounds of the first group can be recognized from the fact that most differences between observed and calculated values are less than k1"C. The S / r ratio is very small (0.010654), proving that the model expresses well the relation.

Table I1 demonstrates the quality of prediction, applying the model to the second group of com- pounds. Most of the residues are within &l"C, and thus the predictability is very satisfactory. The re- sulting R V value measuring the predictive power of the model for the compounds not included in the model is very small (0.5696).

Relations between the average net charges ( in atomic units) of carbon and hydrogen atoms in a molecule with the chain length n ( n 2 10) resulted as

Table I. Statistical Data of the Models

Model

Statistical Both Data Charges Only C' Only 2

R2 0.99835 0.98536 0.99072 S 1.72282 4.93191 3.92549 F 81498.45 2355.19 3737.04 S / r 0.010654 0.030500 0.024276

MELTING POINTS OF OLIGOMETHYLENES 2141

Table 11. Prediction by the Model

Obs. Calc. Resid. No. n ("C) ("C) ("C)

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 61 65 67 69

-25.6 -6.0 10.1 22.1 32.0 40.4 47.4 53.3 60.0 64.0 67.9 71.3 74.7 77.2 80.5 81.7 85.5 87.2

100.3 102.1 104.1 104.7

-24.66 -5.22 10.87 22.30 32.18 40.37 47.79 53.76 59.06 63.88 68.17 71.92 75.33 78.39 81.28 83.60 85.94 88.12

100.67 102.84 103.53 104.64

-0.94 -0.78 -0.77 -0.20 -0.18

0.03 -0.39 -0.46

0.94 0.12

-0.27 -0.62 -0.63 -1.19 -0.78 -1.90 -0.44 -0.92 -0.37 -0.74

0.57 0.06

RV = 0.5696.

@ = -0.014368 + 0.028414/n

+ 0.001572n/( n + 1 ) (14)

( R 2 = 0.99996, S = 3.85*10-6,

significant at 0.01 level)

(15) 2 = 0.006398 - 0.020043/( n + 1)

( R 2 = 0.99998, S = 2.09*10-6,

significant a t 0.01 level).

From eqs. (14) and (15) it is obvious that increase of the chain length decreases @ and hence decreases the difference between @ and - 2 2 . Finally, when n + co, @ and 2 reach the values -0.012796 and 0.006398, respectively (i.e., @ = - 2 2 as demanded by theory). Substituting eqs. (14) and (15) to eq. ( 13) , one obtains

T,,, = 484.4 + 278687.8[ -0.014368 + 0.028414/n

+ 0.001572n/(n + l ) ]

+ 503772.4[0.006398 - 0.020043/(n + l ) ]

= 141.4 + 7918.6/n - 10535.2/(n + 1 ) . (16

2142 TAN AND RODE

PRED~C) T, = TJ1 + ( a - b ) / ( n + b)]

f=tC) -40,m 0,m m,m 120,M)

Figure 1. versus experimental values (RV = 4.9478).

Scatterplot of values predicted by eq. (16)

Generalizing eq. ( 16) , leads to

where Q,, al , a2 and b’ are constants.

gomethylenes [ eq. ( 2 ) ] can be rewritten as The popular equation for melting points of oli-

Table 111. Comparison with Somayajulu’s Work’

= To + T,(u - b ) / ( n + b)

= ah + a L / ( n + b) . (18)

Comparing eqs. ( 17) and ( 18), eq. ( 17) represents eq. (18) with an additional term of a l / n .

The values predicted by eq. (16) for the second group were plotted versus the observed values. The good linear relation between predicted and observed melting points can be seen in Figure 1. From the RV values [0.5696 for eq. (13) and 4.9478 for eq. ( I S ) ] it is obvious, however, that eq. (13) has a bet- ter predictability than eq. ( 16) .

The average net charges of carbon and hydrogen atoms for PE evaluated by eqs. (14) and (15) are -0.012796 and 0.006398, respectively (i.e., @ = -22‘). The melting point of PE predicted by eq. (13) and also by eq. (16) is 141.4’C, in full agree- ment with the experimental value7 also.

The model has satisfactory predictability for both short-chain and long-chain and, in particular, for odd-C and even-C chain oligomethylenes as well, especially when n < 20. The predictability of the model was compared with that of Somayajulu’s study,* as illustrated in Table 111. In shorter oligo- methylenes, the charge distribution shows appar-

OBS Somayajulu’s Study Our Study

n T, T,(n) - T,(n - 1) Calc. Resid. T,(n) - T,(n - 1) Calc. Resid. T,(n) - T,(n - 1)

6 -95.4 -126.5 31.1 -103.6 8.2 7 -90.6 4.8 -96.6 6.0 29.9 -96.5 5.9 7.1 8 -56.8 33.8 -73.3 16.5 23.3 -58.1 1.3 38.4 9 -53.6 3.2 -54.6 1.0 18.7 -51.8 -1.8 6.3

10 -29.7 23.9 -39.2 9.5 15.4 -26.9 -2.8 24.9 11 -25.6 4.1 -26.2 0.4 13.0 -24.7 -0.9 2.2 12 -9.7 15.9 -15.1 5.4 11.1 -10.8 1.1 13.9 13 -6.0 3.7 -4.5 1.5 10.6 -5.2 -0.8 5.6 14 5.5 11.5 2.9 1.6 7.4 4.5 1.0 9.7 15 10.1 4.6 10.2 -0.1 7.3 10.9 -0.8 6.4 16 18.1 8.0 16.8 1.3 6.6 17.7 0.4 6.8 17 22.1 4.0 22.6 -0.5 5.8 22.3 -0.2 4.6 18 28.2 6.1 27.9 0.3 5.3 27.3 0.9 5.0 19 32.0 3.8 32.7 -0.7 4.8 32.2 -0.2 4.9 20 36.7 4.7 37.1 -0.4 4.4 35.9 0.8 3.7

335 134.9 131.9 3.0 133.7 1.2 377.8 136.1 132.9 3.2 134.6 1.7 515.3 135.9 135.0 0.9 136.4 -0.5 912.3 138.0 137.6 0.4 138.5 -0.5

3863.2 139.4 140.4 -1.0 140.7 -1.3 6001.4 140.4 140.8 -0.4 141.0 -0.6

MELTING POINTS OF OLIGOMETHYLENES 2143

ently different characteristics for odd and even numbers of carbons. This characteristic charge dis- tribution naturally leads to different intrachain in- teractions and is reflected in the melting points. The explicit use of the charges in our model is, therefore, the reason for the better predictability of the model for odd-C and even-C chain oligomethylenes com- pared to previous studies. However, it is observed that the difference of charge distribution between them decreases with the increase of chain length and when n > 20, there is no difference observed.

CONCLUSION

The established model for melting points of oligo- methylenes based on quantum chemically calculated net charges has an excellent predictive power, also for compounds not included in the model. The model indicates that the physical properties of oligometh- ylenes are strongly shaped by their electronic struc- ture, which can be simply represented by atomic net charges. Although these net charges depend to some extent on the chain length n, n as sole descriptor gives less satisfactory statistical characteristics in melting point prediction. Since the quantum chem- ical data will also be able to reflect substitution ef- fects, 2o quantitative electronic structure-property relationships appear as a promising method to deal with properties of a variety of polymer compounds.

A grant from the Austrian Federal Government for T. T. M. Tan is gratefully acknowledged.

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Received July 4, 1995 Revised April 18, 1996 Accepted April 19, 1996