The Estimation of Mechanical Properties of Polymers from Molecular Structure ppt

ESTIMATION OF MECHANICAL PROPERTIES OF POLYMERS 1333 volume and the molar volume of polymers.'-3 Van der Waals volume is defined as the space occupied by a molecule that is impenetrable

The Estimation of Mechanical Properties of Polymers

from Molecular Structure

J T SElTZ

The Dow Chemical Co., Central Research, 1702 Building, Midland, Michigan 48674

SYNOPSIS

The use of semiempirical and empirical relationships have been developed to estimate the

mechanical properties of polymeric materials Based on these relationships, properties can

be calculated from only five basic molecular properties They are the molecular weight, van

der Waals volume, the length and number of rotational bonds in the repeat unit, as well

as the Tg of the polymer Since these are fundamental molecular properties, they can be

obtained from either purely theoretical calculations or from group contributions The use

of these techniques by polymer chemists can provide a screening technique that will sig-

nificantly diminish their bench time so that they may pursue more creatively the design

of new polymeric materials 0 1993 John Wiley & Sons, Inc

The purpose of this paper was to give polymer

chemists a technique for estimating the important

mechanical properties of a material from its molec-

ular structure Hopefully, this will provide a screen-

ing tool that will significantly diminish their bench

time so that they may pursue more creatively the

design of new polymeric materials

The important practical applications of polymers

are generally determined by a combination of heat

resistance, stiffness, strength, and cost-in short,

the engineering properties of a material Other

properties may be of importance, but, if a polymer

does not have a balance of these properties, its

chances for commercial success are very limited To

a large extent, these properties can be associated on

the molecular scale with the cohesive forces, the

molecular size, and the chain mobility The approach

taken here is to relate molecular properties of the

repeat unit to the properties of the polymer Repeat

unit properties can be obtained from group additivity

or by simple calculations

In the usual group contribution approach, little

consideration is given to the association between

molecular properties and macroproperties The re-

Journal of Applied Polymer Science, Vol 49, 1331-1351 (1993)

0 1993 John Wiley & Sons, Inc CCC 0021-8995/93/081331-21

sult is that for each property one wishes to calculate

a new table of fragments must be used One of the purposes of this study was to show that mechanical properties can be estimated from a very few basic molecular properties Thus, we use semiempirical means whenever available to make these associa- tions This has the effect of limiting the number of tables of group contributions necessary to calculate the basic properties, it simplifies the calculation procedures, and it indicates to the theoreticians the approximate form to which their theories may be reduced

Linking the mechanical properties to the molec- ular properties of a material is the equation of state Thermodynamic relationships that involve the pressure, volume, temperature, and internal energy lead to the most fundamental equation of state They are expressed in the following form:

( $)T = (g)" = T a B

Here, U is the internal energy, S is the entopy, and

P , V, and T have their usual meanings, and a = 1 /

V[(dV)/(dT)]p and B = - V [ ( d P ) / ( d V ) ] p are

1331

the thermal expansion and bulk modulus, respec-

tively

For mechanical properties below the glass tran-

sition temperature at constant temperature and very

small deformations, the entropy is assumed to be

constant Above the glass transition temperature (in

the plateau region), the material behaves as a rubber

and the mechanical process can be assumed to be

mostly entropic This leads to the following inter-

esting relationships:

Based on these simplifying assumptions, we will proceed to develop estimates of the mechanical properties of polymers

II PRESSURE-VOLUME-TEMPERATURE RELATIONSHIPS

Below Tg, P = T a B - - ( 3 )

( 4 ) It has been found by a number of investigators that

there is a correspondence between the van der Waals

28.0 56.1 86.1 86.1 84.2 62.5 104.1 100.0 42.1 118.1 192.2 120.0 54.1 114.1 128.2 142.2 170.3 198.4 58.0 72.0 86.0 114.0 138.5 160.0 118.0 100.1 128.2 88.7 254.3 192.2

1.11

1.07 1.20 1.33

2.01 1.44 2.70 2.12 3.83 1.75 2.50 2.13 3.43 2.40 1.62 2.04 2.00 3.09 3.63 4.12 4.40 4.15 2.16 3.03 3.9 3.75 1.45 2.58 1.59 2.80 2.60 2.27 2.65 2.00

5.31 5.86 5.60 5.83 7.61 4.85 5.50 4.90 8.50 5.40 4.42 5.13 7.05 5.40 5.75 6.05 6.80 6.00 6.45 7.26 7.26 6.60 4.97 5.90 3.78 6.10 6.00 4.87 5.35 4.55

a Ref 5 All the densities reported from this reference are cited at the glass transition temperature

Ref 6

Internal data of The Dow Chemical Co

ESTIMATION OF MECHANICAL PROPERTIES OF POLYMERS 1333

volume and the molar volume of polymers.'-3 Van

der Waals volume is defined as the space occupied

by a molecule that is impenetrable to other mole-

cules.' Van der Waals radii can be obtained from

gas-phase data4 and bond lengths can be obtained

from X-ray diffraction studies Using these data, the

volume may be calculated for a particular molecule

Bondi' and Slonimskii et a1.2 calculated group con-

tributions to the van der Waals volume for large

molecules and demonstrated the additivity

Since polymers consist of long chains, which

dominate their configuration as they solidify into a

glass, one might expect that they would pack quite

similarly regardless of their quite different chemical

natures To determine if this hypothesis is correct,

it is necessary to obtain the molar volume a t some

point where the polymers may be expected to be in

the same equivalent state and to compare them with

a measurable molecular volume such as the van der

Waals volume Two temperatures are of interest:

absolute zero and the glass transition temperature

At absolute zero, all thermal motion stops and the

material is in a static state The glass transition is

considered to be the point where the material begins

to take on long-range motion and the properties are

no longer controlled by short-range interactions

In Table I, we have compiled the densities and

the thermal coefficient, in terms of the slope of the

volume-temperature curve, from several sources in

the literature We have then calculated the volume

at the glass transition temperature and a t 0 K using

a straight-line extrapolation of the data The results

are tabulated in Table 11 The data from Table I1 is

then plotted as van der Waals volume vs the molar

volumes and fit with a straight line that was forced

through zero The results of these plots are shown

in Figure l ( a ) - ( c )

It is apparent from the data that there is a rea-

sonably good fit between the molar volumes a t the

selected equivalent states To determine the validity

of the approximation, thermal expansion data rang-

ing from room temperature down to 14 K were ob-

tained from the work of Roe and Simha.7 A fifth-

degree polynomial was fit to the data (see Fig 2 )

and the volume-temperature curves were then ex-

tracted from the data by using eqs ( 7 ) and (8) :

PS PMMA

PP PaMS PET PDMPO PBD PEMA PPMA PBMA PHMA POMA PVME PVEE PVBE PVHE PCLST PTBS PVT PEA PBA SAN 76/24

PC PEIS

20.5 40.9 45.9 45.9 61.4 29.2 62.9 56.1 30.7 73.1 94.2 69.3 37.4 66.3 76.6 86.8 107.3 127.7 34.4 44.6 54.8 75.3 72.7 104.7 74.0 56.1 76.5 53.8 136.2 94.2

28.9 60.1 69.4 72.7 100.5 45.4 100.9 86.7 47.5 115.2 147.6 116.4 44.1 102.6 119.2 134.5 165.2 184.4 54.0 72.2 87.4 115.6 143.3 171.7 117.3 88.3 109.0 84.6 220.3 145.0

28.2 58.4 62.9 67.2 90.7 38.7 91.2 78.6 44.1 102.3 137.1 104.6 42.0 90.7 104.9 117.5 145.2 165.0 50.7 67.2 80.1 107.1 135.5 155.0 110.0 81.4 102.1 76.9 197.8 132.5

26.9 53.6 55.9 57.5 81.4 35.1 79.6 68.2 38.3 86.4 119.0 86.9 36.9 81.9 96.7 109.6 134.5 168.7 44.3 60.1 73.8 100.7 116.6 133.4 100.0 73.0 91.7 68.0 162.8 116.6

where a , b , c , d , e , and f a r e coefficients from the

fifth-degree polynomial fit, and T = temperature, K

The thermal expansion curves show very clearly the various transitions due to thermally activated molecular motions However, when these data are integrated to give the volume-temperature curves, these transitions are smeared out into what appears

to be a nearly continuous function as can be seen

in Figure 3 ( a ) - ( c ) The results can be fit with a T1.5 relationship as predicted by free-volume theory? However, from 150

K to the glass transition temperature, the data can

be very nicely approximated by a straight line These relationships are shown by the solid and dashed lines

in Figure 3 ( a ) - ( c ) and are described by eqs ( 10) and ( 11) Table I11 summarizes the data for the six different materials:

rp 1.5

SloDe = 1.42

' Std: dev = 7.84 Correlation index = 0.995

0

van der Waals Volume, cc/mol

van der Waals Volume, cdmol

Figure 1 ( a ) Van der Waals volume vs molar volume at the glass transition temperature;

( b ) van der Waals volume vs molar volume for the glass at 0 K; ( c ) van der Waals volume

vs molar volume of the rubber a t 0 K

T

T8

where T, = glass transition temperature, K, and 6

= V, - Vog = 0.15

Based on the results from these data, we feel jus-

tified in defining the slope of the volume-tempera-

ture curve as a constant over a wide range of tem- peratures This approximation allows the data to be described by the Simha-Boyerg-type diagram as shown in Figure 4 Further, the volume can be de- scribed as being distributed in three parts: (1) the van der Waals volume or the volume considered to

be impenetrable by other molecules; ( 2 ) the packing

v = 6 - + vog ( l1 )

ESTIMATION OF MECHANICAL PROPERTIES OF POLYMERS 1335

Figure 2 Thermal expansion data of Roe and Simha7 fit with a fifth-degree polynomial

volume that reflects the shape and long-chain nature

of the molecule; and ( 3 ) the expansion volume that

is due to thermal motion of the molecules

Using the values generated from the straight-line

fit of the data in Figure 1 ( a ) - ( c ) , the slope and the

intercept of the volume-temperature curve can be

established for amorphous polymers in both the

glassy and rubbery state:

The thermal expansion coefficient can thus be ob-

tained by differentiating eqs 1 2 ( a ) and (12b) and

by using its standard definition:

1

(13b)

1 d V

The density can be estimated from the molecular

weight of the repeat unit divided by the molar vol-

ume:

M P=v

B Pressure-Volume

The pressure-volume-temperature response in polymers can be determined by several molecular factors They include intermolecular potential, bond rotational energies bond, and bond-angle distortion energies The bond-angle distortion energies are important in anisotropic systems where aligned chains are subjected to a stress or pressure In iso- tropic glasses where the bonds are randomly ori- ented, the properties are controlled by rotational and intermolecular potentials In the following sections,

we will separate these into entropy and internal en- ergy terms and then try to relate this to the molec- ular structure using properties that can be related

to the molecular structure either by direct calcula- tion or through quantitative structure property re- lationships (QSPR)

In a perfect crystalline lattice, specific short-range and long-range interactions can be accounted for, but amorphous polymers by their very nature do not fit these computational schemes Several attempts have been made a t using quasi-lattice models to de- scribe the equation of state.1°-16 Most of these are quite limited and need additional information about reduced variables or lattice types Computer models using molecular mechanics techniques have been devised based either on an amorphous cell, which is generated from a parent chain whose conformation

is generated using rotational isomeric-state calcu- lations, l7 or on computer models that also start from RIS configurations and generate radial distribution functions." Both approaches use an l / r 6 potential function to calculate the state properties

ESTIMATION OF MECHANICAL PROPERTIES OF POLYMERS 1337

Here, we will divide the polymer molecule into

suitable submolecules (repeat units) that will then

be assumed to be surrounded by a mean field at a

distance r We will also assume that the volume of

the repeat unit can be described in terms of its van

der Waals volume The field potential will be de-

scribed by a Lennard-Jones" potential function:

Thus, the molar volume is related to the intermo-

lecular distance r as follows:

Nr

C

v = -

where N is Avogadro's number and C is a constant

that corrects for the geometry of the submolecule

On substituting the ratio of the volumes, one arrives

a t the following relationship between volume and

U = Ne and at the minimum Uo = Neo (18)

Equations (15) and (18) combine to define the con-

tribution to the internal energy U from the inter-

molecular potential:

Taking the partial of U with respect to V and sub-

stituting into the thermodynamic equation state for

P below the glass transition temperature yields

p = ( g)T - y [ (; )' - (+)'I ( 20)

At zero pressure and constant temperature,

where Vis the molar volume, cc/mol; V,, the molar

volume a t the temperature of interest, cc/mol; Vo, the molar volume a t the minimum in the potential well; and Uo, is the depth of the well

The bulk modulus is defined by -V [ ( d P ) /

(dV)]T Taking the derivative of eq (22) and mul- tiplying by V gives

Figure 4 Volume-temperature diagram

0

B = ? [ ( Y ) 5vo -(?)'I ( 2 3 )

Haward2' used a form of this equation to predict

the relationship between the volume and the bulk

modulus of poly ( methyl methacrylate)

Pressure-volume data were obtained from

Kaelble'l was fit to eq ( 2 3 ) using regression analysis

to solve for the value of Uo The factor Vo was as-

sumed to be the molar volume of the glass at 0 K

( 1.42 V,,,) The solid line shows the fit to the data in

Molar Volume, cc/mol

Figure 5 ( a ) Pressure-volume dataz1 for polystyrene;

( b ) pressure-volume dataz1 for poly ( methyl methacry-

late ) ; (c ) pressure-volume data21 for poly (vinyl chloride)

Table IV Volume Data

Uo as Calculated from the Pressure-

Table IV shows the values of Uo that were ob-

tained from the fit of the data along with the molar cohesive energy as calculated from the data of Fe- dors22 and van der Waals volume from Bondi' and

Slonimski et a1.2 as compiled in Ref 23 The ratio

of Uo to the cohesive energy for these three polymers

averages 1.94, or approximately 2 We will show later

from the analysis of the mechanical properties that

this ratio, Uo/ Ucoh, is indeed very close to 2

into the calculations as volume changes by using the

ESTIMATION OF MECHANICAL PROPERTIES OF POLYMERS 1339

where the subscripts x , y , and z denote the stresses

and strains in three principal directions

In the case of a material being stretched uniaxially

in the x direction, eq ( 2 5 ) can be solved for the

strain in terms of Poisson's ratio and the volume by

using eq ( 2 7 ) :

e = J d e = J v, ( 1 - 2v)V d V (28)

1

V

Using the results of eqs ( 2 6 ) and ( 2 8 ) with the

assumption that the stress is zero in all directions,

except in the x direction, eq (22) can now be solved

in terms of stress and strain where V, is the volume-

dependent strain:

Similarly, substituting eqs ( 24) and ( 2 8 ) into eq

( 23 ) , the tensile modulus is obtained

E = 24(1 - 2v)Ucoh [ 5- ; - 3 - :] (30)

The value of V, can be estimated from van der

Waals volume and the glass transition temperature

using eqs ( l o ) , ( l l ) , or ( 1 2 ) , and Uo may be esti-

mated using the approximation that it is two times

the cohesive energy However, without a relationship

between Poisson's ratio and the molecular structure,

we are unable to calculate the tensile or shear

moduli

2 Poisson's Ratio

Our model as presently constituted does not contain

any information about the directional properties of

the system However, just as the bulk modulus varies

as 1 / V in eq ( 2 3 ) , one might expect that a simplified

unidirectional (tensile) moduli would vary as the

area being stressed Using this analogy, Poisson's

ratio can be equated to E / B from eq ( 2 5 ) The

result of this relationship can be stated as follows:

v = 0.5 - k f i (31)

(32)

where 1, is the length of the repeat unit in its fully extended conformation and NA is Avogadro's num- ber The fully extended conformation corresponds

to the all-trans-conformation and can be calculated

by assuming ideal tetrahedral bonding around the carbon atoms in the polymer backbone and using simple trigonomeric relationships Table V gives the

calculated A for a number of polymers for which we

have data

The data from Table V is plotted in Figure 6 and

is represented by the circular symbols The line in Figure 6 was obtained by fitting the data to a square- root argument using regression analysis The sta-

tistical fit represented by eq (33) has a standard

deviation of 0.019 and a correlation index of 0.998:

(33)

To estimate the stress-strain relationship as a func- tion of temperature, we must have both Poisson's ratio and the volume as a function of temperature The temperature-volume relationships can be cal- culated from eq ( 12a)

A = - v w

NA 1,

v = -2.37 X 1 0 6 f i + 0.513

Table V Cross-sectional Area

Poisson's Ratio and Molecular

M o 1 e c u 1 a r Cross-sectional Poisson's Area Polymer Ratio (cm2 x 1 0 - l ~ ) Polycarbonate

PS ST/MMA 35/65 Poly(p-methyl styrene) SAN 76/24

PSF PDMPO PET POMS Arylate 1 : 1 : 2 Phenoxy resin PMMA PTBS PVC Poly(amide-imide)

The value A can be thought of as the molecular

cross-sectional area and is defined here as the area

of the end of a cylinder whose volume is equal to

the van der Waals volume of the repeat unit and

has a length of the repeat unit in its all transcon-

figuration:

a Internal data of The Dow Chemical Co Poisson's ratio was measured using an MTS biaxial extensometer no 632.85B-05 in

conjunction with an MTS 880 hydraulic testing machine The

tests were performed under the conditions of ASTM D638, using type 1 tensile specimens The crosshead speed was 0.2 in./min All samples were compression-molded and then annealed at (T,

30 K) for 24 h

Data obtained from Ref 24

Poisson's ratio increases very slowly as a function

of temperature to within 20" of the glass transition

temperature with only very minor deviations due to

low-temperature transitions Just above the glass

transition temperature, Poisson's ratio is assumed

to approach 0.5 so that the volume is conserved in

the rubbery state Based on the data and the previous assumption, a fitting function was developed to es-

Table VI Poisson's Ratio as a Function of Temperature