1.14 Molecular Dimensions in the Amorphous State
1.14.2 Root Mean Square End-to-End Distance of Flexible Macromolecules
1.14.2.1 Freely Oriented Chains
The simplest calculation is based on the assumption that a macromolecule com- prisesσ11 (sigma11) elements of equivalent size which are joined byσbonds of fixed length l. (If the bonds differ in length, an average value can be used in this calculation. Here we assume that all bonds are equivalent.) All angles between successive bonds are equally probable. Such a chain is illustrated in Fig. 1.8where each bond is represented by a vectorli. The end-to-end vector in a given conformation is
d5Xσ
i51
li (1-24)
Now it is convenient to recall the meaning of the dot product of two vectors.
For vectors a and b, the dot (or scalar) product is equal to the product of their lengths and the cosine of the angle between them. That is,
ab5ab cosθ (1-25)
whereθis the bond angle (which is the supplement of the valence angle, for C—
C bonds) anda andb are the respective bond lengths. The dot product is a scalar
quantity and the dot product of a vector with itself is just the square of the length of the vector. To obtain the end-to-end scalar distance, we take the dot product of d. Because the single chain can take any of an infinite number of conformations, we will compute the average scalar magnitude ofd over all possible conforma- tions. That is,
hd2i5hd di5 Xσ
i51
li
! U Xσ
j51
lj
!
* +
(1-26)
The dummy subscripts i and j indicate that each term in the second sum is multiplied by each term in the first sum.
If θis the angle between the positive directions of any two successive bonds, then
lili115lili11 cosθ (1-25a) All values ofθare equally probable for a chain with unrestricted rotation, and since cosθ5 2cos(2θ)
lili115lili11hcosθi50 (1-27) l15
l17 l14
l13 l18
l19 l20 l11
l6
l9
l10 l4
l5 l8
l29
l27
l26
l25
l24 l23 l22
θ
d l7
l3
l2
l1
l28 l12
FIGURE 1.8
An unrestricted macromolecule. The bond lengthsliare fixed and equal and there are no preferred bond angles.
49 1.14 Molecular Dimensions in the Amorphous State
Thus all dot products in Eq. (1-26) vanish except when each vector is multi- plied with itself. We are left with
hd2i5 Xσ
i51
l2i
* +
5σl2 (1-28)
The extended or contour length of the freely oriented macromolecule will be σ l. Its root mean square (rms) end-to-end distance will be lσ1/2 from the above equation. It can be seen that the ratio of the average end-to-end separation to the extended length is lσ1/2. Sinceσwill be of the order of a few hundred even for moderately sized macromolecules,dwill be on the average much smaller than the chain end separation in the fully extended conformation.
The average chain end separation which has been calculated gives little infor- mation about the magnitudes of this distance for a number of macromolecules at any instant. When this distribution of end-to-end distances is calculated, it is found, not surprisingly, that it is very improbable that the two ends of a linear molecule will be very close or very far from each other. It can also be shown that the density of chain segments is greatest near the center of a macromolecule and decreases toward the outside of the random coil.
For real polymer chains, there exist local correlations between bond vectors with restricted bond angles and steric hindrance (see more sophisticated models described below). Therefore, the actual mean square end-to-end distance of real polymer chains must be larger than the one calculated using the freely oriented chain model. Flory introduced the concept of characteristic ratio to signify such difference. The characteristic ratio is chain length dependent when σ,100.
However, for infinite long chains, the characteristic ratio (i.e.,CN) is given by
CN5 d2 real=σl2 (1-29)
The numerical value of CN depends on the flexibility of the polymer chain.
Here,CNis always greater than 1 and, for flexible polymers, typical values vary from 5 to 10. For example, at 140C, the characteristic ratio of polyethylene is 6.8 while that of poly(methyl methacrylate) is 9.0.
1.14.2.2 Freely Rotating Chains
The random-flight model used in the previous section underestimates the true dimensions of polymer molecules, because it ignores restrictions to completely free orientation resulting from fixed valence bond angles and steric effects. It also fails to allow for the long-range effects that result from the inability of two seg- ments of the chain to occupy the same space at the same time.
The effects of fixed rather than unrestricted bond angles can be readily com- puted, and it is found that the random flight relation ofEq. (1-28)is modified to
hd2i5σl2ð11cosθÞ=ð12cosθÞ (1-30)
EXAMPLE 1-1
For a polyethylene chain,lis the C—C bond distance (1.54310210m or 0.154 nm) andθ is 180minus the tetrahedral bond angle51802109280570 320. Cosθthen is 0.33 and hd2i52σ l2. The chain end-to-end distance calculated using this model is thus expanded by a factor of ffiffiffi
p2
for this reason. And the correspondingCNvalue is 2.
1.14.2.3 Hindered Rotation Chains
When some conformations are preferred over others (e.g., inFig. 1.6), the chain dimensions are further expanded over those calculated, andEq. (1-29)becomes
hd2i5σl2 11cosθ 12cosθ
11hcosφi 12hcosφi
(1-31)
wherehcos φi is the statistical mechanical (not arithmetic) average value of the cosine of the rotation angleφwith probabilities determined by Boltzmann factors, exp(2E(φ)/kT). The effect of Boltzmann factors is that rotation angles with high energy contribute less to the average as such rotation angles do not occur fre- quently, especially at low temperatures. For free rotation, all values of φ are equally probable, cos φ is zero, and Eq. (1-31) reduces to Eq. (1-30). In a completely planar zigzag conformation, all rotamers are trans and φ50. Then hcosφi51, and the model breaks down in this limit. Nevertheless, it does show that the chain becomes more and more extended the closer the rotation angle is to zero. The values of hcos φi can be calculated if the functional dependence of potential energy of a sequence of bonds on the bond angle is known. For small molecules, this can be deduced from infrared spectra.Figure 1.6showed this rela- tion approximately for a normal paraffin.
The value of hcos φi will depend on temperature (Boltzmann factors), of course, since the molecule will have sufficient torsional motion to overcome the energy barriers hindering rotation when the temperature is sufficiently high.
EXAMPLE 1-2
To illustrate how to calculate hcos φi using Boltzmann factors, let’s assume that each CaC bond in a polyethylene molecule obeys the torsional potential curve shown inFig. 1.6 (i.e., each bond is either in thetrans,g2, org1 state, not in any other rotational angles).
Given that at 140C (413 K), the difference between the torsional potentials between the transandgauchestates (i.e.,Δe) is about 2100 J/mol. Fraction of the bonds in thetrans state is given by
51 1.14 Molecular Dimensions in the Amorphous State
exp2 0 R3413
exp2 0
R3413
12 exp2 2;100 R3413 50:48
Here, R is the universal gas constant (8:314 molJK). Therefore, the fraction of the bonds in either one of the gauche states is (120.48)/250.26. And hcos φi50.48 cos (0)10.26 cos (120)10.26 cos (240)50.22. Using the calcu- lated hcos φi and the corresponding θ (70 320) in Eq. (1-31), the chain end- to-end distance at 140C is thus expanded by a factor of ffiffiffi
p3
for this reason. And the corresponding CNvalue is 3. It is clear that by incorporating more structural details of the monomer into the calculation of CN, its value becomes closer to that of the experimental values (i.e., 6.8 at 140C).
1.14.2.4 Rotational Isomeric State Model[9]
In the hindered rotation chains, the barriers to the rotational motion of individual bonds are assumed to be independent. However, in the rotational isomeric state model, such barriers are considered to be dependent for two consecutive bonds.
For example, in the case of polyethylene, a consecutive sequence of two bonds in the g1 andg2 has high energy, thereby a low probability to occur while two consecutive bonds withg1 andg1, a relatively lower energy combination, are more probable. A full description of the rotational isomeric state chains is beyond the scope of this textbook. Interested readers should refer to reference[9] for fur- ther details. Nevertheless, the rotational isomeric state model has been incorpo- rated into computer programs along with interatomic potentials, bond angles, and so on to model the lowest energy conformations of macromolecules in specified environments. Such molecular simulation studies have shown that the lowest energy state of polymers in their “melt” condition is not necessarily that with the highest entropy. In particular, molten polyethylene molecules do not resemble a bowl of spaghetti. Rather, the overall conformation with the lowest energy is one that comprises a significant fraction of shapes in which the chains are folded back on themselves in an expanded version of the polyethylene crystal structure described in Chapter 4.
When the average end-to-end distance of a macromolecular coil is described by the rotational isomeric state model, the polymer is said to be in its “unperturbed”
state. Its dimensions then are determined only by the characteristics of the molecule itself (i.e., bond lengths, bond angles, and barriers of rotation angles), not the inter- atomic potentials. In general, the end-to-end distance of a dissolved macromolecule is greater than that in its unperturbed state because the polymer coil is swollen by solvent. If the actual average end-to-end distance in solution ishd2i1/2, then
hd2i1=25hd20i1=2 α (1-32)
where the subscript zero refers to the unperturbed dimensions. The expansion coefficient α can be considered to be practically equal to the coefficient αη, which will be introduced inSection 3.3in connection with the ratio of intrinsic viscosities of a particular polymer in a good solvent and under theta conditions.
If a random coil polymer is strongly solvated in a particular solvent, the molecu- lar dimensions will be relatively expanded andα will be large. Conversely, in a very poor solventαcan be reduced to a value of 1. This corresponds to theta con- ditions under which the end-to-end distance is the same as it would be in bulk polymer at the same temperature (Section 3.1.4).
1.14.2.5 The Equivalent Random Chains[10]
The real polymer chain may be usefully approximated for some purposes by an equivalent freely oriented (random) chain. It is obviously possible to find a ran- domly oriented model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will fit this requirement and the additional stipulation that the real and phantom chains also have the same contour length.
If both chains have the same end-to-end distance, then
hd2i5hde2i5σel2e (1-33) where the unsubscripted term refers to the real chain and the subscript e desig- nates the equivalent random chain. Here,leis usually referred to as Kuhn length.
Also, if both have the same contour lengthD, then
D5De5σele (1-34)
FromEqs. (1-33) and (1-34):
le5hd2i=D (1-35)
and
σe5D2=hd2i (1-36)
EXAMPLE 1-3
Calculate the Kuhn length of polyethylene at 140C. At 140C,CN56.8. Andl50.154 nm andθ570320. Here,le5hd2i/D5CNσl2/D, whereD5σlcos(θ/2). Therefore,le5CNl/cos (θ/2)51.3 nm.
53 1.14 Molecular Dimensions in the Amorphous State