F. Rodriguez Cornell University ithaca. New York 14853
II
Tape Analogs for Long Chain
The dependence of distance r between two atoms in a chain separated by n bonds can be derived mathematically for the cases of the freely jointed chain and the chain with constant bond angle. When restrictions on rotational states are introduced, the problem becomes complex. The excluded volume restriction is very difficult to incorporate even in a computer-aided solution. I t is shown here that a small number of models made by bending aluminum tape in a special jig permits reasonable illustration of all these effects. A erouo - . of 16 students. each soendine" about two hours constructing and making measurements on individual models, generated sufficient information to show the dramatic increase in r2 due to restriction on rotational states and excluded volume. The finished models enable the students t o visualize the statistical nature of chain conformations in three dimensions. The picture of a polymer as a long sequence of hifunctional units connected to give a thread-like structure is a useful one. Of course, some highly cross-linked materials such as epoxy or phenolic resins and other thermosets do not fit the picture. However, almost all the molecules comprising fibers, rubber, thermoplastics, and lacquers, as well as the naturally occurring proteins and polysaccharides are linear. In the process of being put into useful form, or even in the final form in the case of rubber. the "linear" oolvmer . is in an amorphous, isotropic state sometimes referred to as a random coil. The terminology is a hit unfortunate since the word coil implies a spiral-like regularity whereas the true oicture is a chaotic one with verv little order oersistine beyond the covalent bonds attached t o each atom: In the random state, a parameter which describes the space occupied by the polymer molecule (in dilute solution or melt) or a polymer segment (in a cross-linked rubber) is the distance, r , between two atoms of the same chain separated by n honds, each of length a. For many purposes, the average value of the square of r is all that is required. If the bonds follow one another with no constraints as to preferred direction, we have a "freely-jointed chain" corresponding to a random flight. In this case, the root-meansquare of the end-to-end distance is given by ( I )
.
-
(r2)1/2/a = &2
(1)
Real polymer chains have successive honds connected to each other a t fixed angles. If the angle between successive honds is 0', and 8 = 1800 - Of, then the expression for random flight is modified ( I ) (r2)'I2/o= n1I2tan(t'l2) = [n(l + cost)/(l -cost)]'"
(2)
For the tetrahedral angle typical of carbon chains, 8 = 70.5O and cos0 = 113. Flory considers 0 = 68' a more realistic angle (1). while the concept of r can be dealt with mathematically rather easilv, the phvsical sienificance is not obvious. For example, hb& is it ;elated t o the volume swept out by "coiled up" chain? Physical models can aid in visualizing Presented before the Division of Polymer Chemistry of ACS in Atlantic City, September, 1974. Roovers Tape 2638, Elliot Business Machines, Inc., Randolph, Mass. 02368.
'
the statistical nature of r 2 and also the fluctuations both in the molecular "volume" and in r. Most often, the effect of various intrachain restrictions on the resulting r 2 have heen studied using automatic machine computation. Simplifying assumptions and special mathematical techniaues decrease the burden on the memory of the computer (1-3). In almost all cases, the length of the chain eenerated can be keot small bv introducine the idea of a Gatistical segment. 1; the Nth bond of a chiin is far enough removed from the first hond so that its direction can be considered to he independent of the direction of the first, one can treat a restricted real chain of n bonds as if it were a freely jointed chain made up of rn = nlN segments, each of length ($)1'2, where the latter is the length of a "statistical segment" of N bonds. Once the value of (r.2)'I2 is established subject to various hond rotation angles and hindrances, the result can be applied t o longer chains using eqn. (1) replacing the "a" by the segmental length and n by rn. In order to use the freely jointed chain model, each segment must be long enough so that the direction of the Nth hond is independent of the first bond. This requirement often can be met with N equal to about 10. On the other hand, for (r2) to be proportional to n , a segment of 50-100 honds may be needed (4). Thus for mathematical calculations or physical analogs, "molecules" only 100 honds long need be generated although the real polymers of interest mav be over a hundred times loneer. Non-mathematical analogs or models are quite feasible for regular repeating structures such as individual helices or crystal lattices (5). Models of the chaotic state are not popular since they have to be rather large (50-100 bonds), and since a single specimen does not suffice to illustrate the fluctuating character of the conformation. Kuhn reoorted one of the few instances where chaotic models weie employed not only to obtain dimensions, but also to measure hydrodynamic drag in a real liquid (6). Pieces of wire were divided into 130 segments, each corresponding to about 22 chain atoms in polystyrene. A sphere with a mark was rolled in an arbitrary manner and the wire was bent at each segment end in the direction of the axis of the sphere in its final resting position. Drag experiments in viscous fluids gave reasonable agreement with theories for an expanded bead model with partial immobilization of solvent. Tape Model Method
One of the most useful approximations to decrease the labor of the mathematical aspect of physical model building is that of 'rotational isomeric states." Essentially, we replace the infinite possibilities from free rotation of the third hond (Fig. 1) by three discrete states represented by positions A, B, and C called the trans (t), gauche left (g-), and gauche right (g+), respectively. Although other, more realistic angles are sometimes used, we choose to space these 120° apart. A variety of macroscopic models can be imagined. However, from the standpoint of expense and convenience, aluminum label tape is suitable for models containing as many as 100 honds. I t is stiff enough to retain the shape into which it is bent, hut soft enough to be bent in a simple jig. If the polymer chain is placed in the planar zig-zag conformation (Fig. 2), the tape replaces that portion of the chain
Figure 1. (el A three-bond sequence w l h constant bond angle 8'. The faulm atom is at A. B, or C (b) End view of center band showing bans (A), gauche Ien (8, and gauche right (C) posiuons of third bond.
Figure 3. Bondfanning jig. The Crimping Arm. I, pivots over the Mandrel. 11. The two halves of the Mandrel are sped by two thicknesses of the t a p used In the polymer chain leaving a channel above to hold the tape being bent. Ail dimensions are given in inches based on the use of tape which is % in. wide.
Figure 2. (a) Polymer chain in planar, zig-zag conformation. Shaded strip corresponds to aluminum tape model. (b) Tape model Illustrating appearance of three rotational isomer variations. Figure 4. Bond-forming jig in use. Actual apparatus is brass
containing only the main-chain bonds. The trans state leaves the tape unbent. The gauche conformations involve bending the tape through an angle of f120° (measured in the plane normal to the hinge bond). I t has been found that aluminum tape (0.375 X 0.012 in.)' bent in a heavy, metal jig gives the desirable permanence and accuracy. A consistent bond angle is especially important since a one-degree error in bond angle leads to a 5% error in (r2)'I2. The dimensions of a suitable jig are listed in Figure 3. In Figure 4 one can see the actual chain being generated. The tape is removed from the jig after each bond is decided and turned over so that the next bond will coincide with the bending line. With care, even a cyclohexane ring can he synthesized (g+,g-,g+,g-). There remains the matter of deciding the bond sequence. A simple sequence generator is a single die. Tossing it once for each bond can be a bit tedious, however. One lets the one- and two-spot sides represent t, three- and four-spots represent g+, and five- and six-spots represent g-. Tables of random numbers can serve the same purpose. A novel, and much less tedious chain sequence generator is a mixture of beads or marbles. For example a mixture of beads of three colors (about 40 beads of each) is poured into a trough to form a linear sequence. By pouring about 50 beads at a time, the sequence o f t , g+, g- is deduced a t a glance from the identification of each option with one color. I t is quite simple to vary the relative probability of options by varying the proportions of colors in the mixture. Where certain sequences are forbidden, such as g+g- and g-g+, these are easily removed from the trough before assembly of the model. As Flory has pointed out, the sequence g+g- and its reciprocal g-g+ lead to a very energetically unfavorable situation. As seen in Figure 5, the consequence of chain bond 4 being g+ with respect to bond 2 and hond 5 being g- with
respect to bond 3 is that bonds 1 and 6 will overlap. The overlap is serious whether bonds 1and 6 are continuations of the chain or side-groups. This is the familiar conformation of cyclohexane in the "chair' form. Presentationof Results
In the case of the freely jointed chain, the radial distribution function, P(r) of r is Gaussian and is given by P(r)dr = 4sr2(fl/&P exp (-f12r2)dr
(3)
where P(r)dr is the frequency of finding a chain with an end-to-end distance r terminating in the shell represented by volume 4nr2dr, and 1/f12= (21317
(4)
This assumes a continuous distribution and is restricted to values of r much less than nu. Although this is derived for the freely jointed chain for which eqn.'il) applies, eqns. (3) and (4) are also recommended for use with other values of i2such as those obtained from eqn. (2) as long as the stipulation regarding r heing much less than na is met ( I ) . When the dimensions of a small set of models are to he compared with the expected distributions,.it is convenient to do so on a cumulative basis rather than a differential basis. We need the cumulative probability, CR (the integral of eqn. (3), from r = 0 t o r = R.
If we let t = 2r2O2,we can substitute in eqns. (3) and (5) to get Volume 53. Number 2, February 1976 / 93
Figwe 5. The bond sequence c~espondingto g+gane-like Ebucture.
Asymptote,
leads to a cycbhex-
3 . 4 3 -
/
' o,
/
.
O
A Ten restricfed topes
9
.
mth ranked point out of i total points from the lowest value of y2 is given by
Six random tapes
I
10
n,
20
1
1
50
1
1 1 1 1
KX)
number of bonds
9
Flgure 6. Pooled values of = #/(2na2) for two groups of tape models. For example fl is based an 100 measurements for n = 10 and tan measurements for n = 100 in lhe restricted case. Theory predicts unhy for in the random case. Florq's calculated curve ending with an asymptotic value of 3.43 approximates the experimental rerun for polyelhylene at 140°C.
But
where P(x21v) is the integrated chi square probahility function with v degrees of freedom (7). Identifying parts of eqn. (6) with eqn. (7) we deduce
where x2 = 2R2O2. Experimentally we measure values for R2 and compare them to the expected average. We let a dimensionless ratio be
T o ohtain data, each of six students made one model molecule of 100 honds. This gave six measurements of R for n = 100. Each student measured on his own molecule R from bonds 0-50, 25-75, and 50-100 giving a total of 18 values for n = 50 from the six models. Likewise, there were a total of 42 measurements with n = 20. Despite the relatively small number of data points, the average value of y 2 thus obtained remains between 1.0 and 1.5 (Fig. 6). As n increases, the average is based on fewer and fewer inputs so greater uncertainty is expected. The cumulative distrihution (Fig. 7) is plotted where the experimental CR for the 94 / Journal of ChemicalEducation
9
Figure 7. Individual measurements of = #/? plotted as log on narmei probabilny paper. The dashed line is the same as the sold line except mat it is banslated vertically by a factor ol 3.5.
In using P(X2/3)as the theoretical line (Fig. 7), we are assuming that the distribution of chains with constant bond angle is of the same form as that for the freely jointed case with r 2 multiplied by two ( 8 ) .The values of the integrated chi square probability function are taken from an extensive tabulation (7). The distribution of y2 (Fig. 7) is displaced to higher values from the theoretical predictions corresponding t o the higher average value in Figure 6. The general slope of the distribution for n = 50 and n = 100 is quite the same as predicted. In the same project, ten other students made models in which two restrictions were introduced. I t has heen noted earlier that the sequence g+g- and its reciprocal g-g+ are energetically unfavorable conformations. In computer calculations a low probability can be assigned to this sequence. For our purpose, it was convenient to set the probability a t zero. Additionally, the closest approach allowed between sections of the polymer chain was 1in. (about two C-H bond lengths). The average values of y2 are much higher (Fig. 6) and the distribution of y2 (n = 100) is altered (Fig. 7). I t is especially obvious that the smaller values of y 2 become unlikely under the imposed conditions. Flory's values for y2 (identified as C,/2 in his nomenclature) were obtained using realistic values for rotational energy barriers, bond angle, and rotation angle. Since his results approximate the experimental values for polyethylene a t 140°C, it is worth noting that the present averages indicate a proper trend despite the small numher of models built (Fig. 6). The tape model used here is equivalent to a random flight with or without certain restrictions in a lattice. Rather than use the analog, a student may wish to use a computer to generate molecules. The g+g- restriction can be put in even a primitive program. However, the long range excluded volume effects are very difficult to program without an excessive burden on the memory of the computer. More importantly from the teaching standpoint, the student does not get the picture that the model communicates so readily.
There is a further word of caution in using models t o illustrate concepts. The tape represents a main chain of carbon atoms stripped of the foliage ordinarily attached to each chain atom' The effects of side-grou~s, ly when accompanied by stereoregularity are not visualI,, the area of crystalline polymers mischief has alreadv been done hv "methvlene mvovia" in which the
students of the amorphous state will be able to picture chaotic chains which bear methyl, phenyl, and pyrrolidyl
groups as well as the more prosaic hydrogen atoms. Literature CHed (1) Flary, P. J., "Sfatistied Mechanics of Chain Moleculek"
Wile+btemcjenee, New
York. 1969. 12) Volkenatein. M. V.. "Con%urational Statiotk of Polymeric Chains," Wilcy-Intor-
(3)
"'
science. Ncv York. 1963. RUM^, them T ~ c ~46.593 . . (1973).
~..r.J. E..
""an"" New York, 196S.pp. 5.".
(a Flop, P.J..R~I.111,
p. 412.
Volums 53,Number 2. February 1976 /
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