Polymer principles in the undergraduate physical chemistry course

report of the polymer core courfe committee

Polymer Principles in the Undergraduate Physical Chemistry Course Part 2 Physical Chemistry Core Course Subcommittee,' ACS Polymer Education Committee, 1979-1984 The teaching of physical chemistry can and should include a wide range of examples involving polymers. This, the second part of a report prepared by the joint ACS Polymer Education Committee of the Polymer Chemistry and Polymeric Materials Science and Engineering Divisions, covers statistical thermodynamics, conformation, molecular weights, rubber elasticity and viscoelasticity, and the kinetics of polymerization. Also included is an appendix of multiple-choice polymer-chemistry questions. Part 1 covered the application of classical thermodynamics, polymer crystallinity, and phase diagrams to the teaching of physical chemistry ( 1 ) . Statistical Mechanics, Average Values, and Molecular Properties Average Values and Structure In the presentation of statistical mechanics and in the subsequent discussion of molecular properties, some of the principles stressed are the concept of average values, the different methods of averaging, and the use of distrihution functions for this purpose. In Part 1( I ) the formal description of molecular weights and the problem of the molecular weight distribution of chain molecules were mentioned. In the more conventional procedure, one usually calculates the Maxwell-Boltzmann distribution function and from it the different averages of the velocity of ideal gas molecules. If this concept is not generalized the students developed a very limited set of ideas as to the basis for distrihution functions and their potential uses. The same objectives can also be accomplished by studying the properties and structure of chain molecules. This endeavor also serves as a major exercise and generalization as to how the structure of more complex molecules can be determined. The large number of chain atoms that compose a long chain molecule and the obvious impossibility of precisely locating each atom in space, make it appear that the quantitative description of polymer structure would he very difficult, if not impossible. These conclusions would in fact he true if we were limited to the methods developed for the structural analysis

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Journal of Chemical Education

of small molecules. However, the large number of chain atoms involved makes polymers an ideal system for statistical analysis, which greatly simplifies the problem and brings in the use of average quantities. The apparent complexity of the problem can thus he overcome. The thermodynamic functions, as well as other macroscopic properties, are merely averages over the whole system. We have always utilized average values, although never specifically indicating such. Sucha pulyme-r chi~incanrhy its wry nature, assume a large number of different spatial conformations, it lends itself verv naturally to treatment by statistical methods. The most ele". mentary, hut still very informative, analysis is to consider the properties of a hypothetical model representing the real chain. This model is known as a "freely jointed chain" (2). In this model, the bond lengths all are identical, valence angles are ignored, and the direction of any given bond is assumed independent of any other. Although obviously not an accurate

The Physical Chemistry Core Course Subcommittee of the ACS Polymer Education Committee consists of: Rita Blumensteln, University of Lowell, Lowell. MA Charles E. Carraher, Florida Atlantic University. Boca Raton, Howard Coker, University of South Dakota. Vermillion, SD Fredrick Fowkes, Lehigh University, Bethlehem, PA Eckard Hellmuth, University of Missouri, Kansas City, MO David Karl, Wright State University, Dayton, OH Leo Mandelkern, Florida State University, Tallahassee. FL James E. Mark, University of Cincinnati,Cincinnati, OH Wayne Mattice, Louisiana State University. Baton Rouge, LA Ferdinand Rodriguez, Cornell University. Ithaca, NY Charles Rogers, Case Western Reserve University, Cleveland, OH

Leslie Sperling, Chairman, Lehigh University, Bethlehem. PA Richard Stein, University of Massachusetts, Amherst, MA

representation of a real chain, this model allows for a simple calculation that sets up the basic principles of the prohlem and that can be further refined. To analyze this prohlem, we first seek the probability P(r) that the two ends of an individual chainare separated by the distance r. This prohlem is identical to the classical threedimensional random flight whose solution is given by (3) P(r) = ArZeap(-b2r2) (1) where A and b are constants. If r is replaced by u in eqn. (11, then the Maxwell-Boltzmann distribution function for the velocity of gas molecules is ohtained. Hence, one arrives a t exactly the same mathematical form applicable to ideal gases hut from a quite different molecular point of view. This approach should give the student a better feeling for real molecular systems. By usingeqn. (1) one cancalculate, in the usual manner, various average dimensions. For example, although the average value of r is obviously zero the average value of r2 is given by (2) (r2)= 12n where n is the number of links or honds with length 1 in the chain. Thus. we find that a linear dimension of the chain is directly proportional to the square root of the numher of bonds. This is a verv fundamental result. There is an analow ... between this result and the mean square displacement of a gas or to Brownian morion.This tsoe of'ralculation has been eytended to more realistic modeis: I t is always found, no matter how complex the chemical nature of a real chain, that its linear dimension is always proportional to the square root of the numher of honds.

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major features of the helix-coil transition then evolves. A simple treatment of this prohlem, adequate for presentation to an undergraduate course, has been given (5). Molecular Weight Distribution Polvmerization reactions. both svnthetic and natural. generally lead to polymers with a heterogeneous molecular uleizht.. i.e....poI\.rner chains with a difl'erent number of units. ~ o r e c u l aweight r distributions may be relatively broad as in the case for most svnthetic polvmers and manvnaturallv occurring polymers.'lt may be ielatively narrow for ceitain natural polymers (because of the imvosed steric and electronic constraint;), or may he mono-, bi-; tri-, or polyrnodal. A himodal curve is often characteristic of a polymerization occurring under two distinct pathways or environments. Thus, most synthetic polymers and many naturally occurring polymers consist of molecules with different molecular weights and are said to he polydisperse. In contrast, specific proteins and nucleic acids, like typical small molecules, consist of molecules with a specific molecular weight (M) and are said to be monodisnerse. Sc\waI mathmxnical momt:nts can In, descritxd using the differential or frcournw dis~rihutioncurve. and can he described by equations. Tbe first_moment is called the number-average molecular weight. M,. Anv measurement that leads to thk numher of mol&ules, functional groups or particles, that are plesent in a given weight of sample, allows &e calculation of M,. The number-average molecular weight M,, is calculated like any other numerical average by dividing the sum of the individual molecular weight values by the number of molecules. This solution is shown mathematically:

Confiaurational Partition Function

In discussing the partition function we traditionally a i sume that there are three contributions, the translational. vibrational, and rotational partition function. This partition function can he formulated in a straightforward way using the methods of matrix algebra (4). Most students in the firstyear course are equipped to handle this algebra. These methods are very often used in the discussion of quantum chemistry. The most complex matrix involved is a 3 X 3; most problems can be demonstrated with matrices no large than 2 X 2. The formulation of the partition function of a longchain molecule is a triumuh of statistical mechanics when the large number of formakons available to even a modestsize chain is recognized. The configurational partition function is proportional to the largest eigenvalue of the matrix of statistical weights. This matrix is constructed by representing the statistical weight of each of the accessible rotational states. The development of this partition function also introduces the studentto the methdds of conformational analysis. The major consideration with respect to the accessibility of states, and the associated statistical weights is the potential hindering rotation about the internal bonds. The prohlems in this regard for a long chain molecule, such as polyethylene for example, is no different from those for an nalkane. The formulation of the partition function enables a determination to he made of a varietv of average These - vro~erties. . . include the mean square end-&end distance, the average dipole moment as well as other quantities. The mean square end-to-end distance of more complex chains can be calculated in a very rigorous, mathematically exact manner by these methods. Chain models no longer need to he involved to represent real molecules. Another application of these matrix methods allows for a straightforward treatment of order-disorder phenomena such as the one-dimensional helix-coil transition (5, 6). Here a chain is represented by either the helical or random coil (disordered) states. The appropriate partition function can be formulated when account is taken of the energy required to initiate the helical sequence and for its propagation. The

where Mj is the molecular weight of the N;molecules. Collieative uronerties deneodent on the numher of narticles . present are obviously related to R.. Them,values are indeoendent of molecular size and are hiehlv " .sensitive to small molecules present in the mixture. Values for %, are determined hv Raoult's techniaues that are deuendent on colliaative properties such as ebulliometry (boiling-point elevation), cryometry (freezing-point depression), osmometry, and end-group analysis. Weight-average molecular.weight, X-, is determined from experiments in which each molecule or chain makes a contribution to the measured result. This averaae is more dependent on the number of heavier molecules than is the number-avera~emolecule weight, which is dependent simply . on the numbe; of particles. The weight-average molecular weight is the second moment or second power average as shown mathematically

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~

Bulk nronerties associated with laree deformations. such as . . \,isrosity nnd toughn+:is,;in.prt~cul;irlydftcted hv ;ii.values. G, values are determined hv light-sratterine and ultracentrif"gation techniques. ~ i ~ h moments k r canuhe defined followina the scheme of ems. (4) and (5). For examvle, we define tce Z average mo1ec"lar weight, Mz as

The lower moments or averaaes can be determined exoerimentally. The numher average can be ohtained from the colligative type measurement. Therefore, in the expression for the osmotic pressure, eqn. (lo), the molecula; weight should be replaced by M.. A colligative property counts the number of molecules. The weight average molecular weight can be determined from light-scattering experiments or by equilibrium ultracentrifugation. The ultracentrifuge can Volume 62

Number 11 November 1965

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also he used to determine Mr. These are absolute methods for determining molecular wiight which have their origin in thermodynamic analysis. Hence the study of macromoleculesgeneralizes the concept of molecular &eights and introduces the very important concept of molecular weight averages. There are numerous methods employed in determining the molecular weight of a compound. Only three of these are useful in determining the molecular weight of polymers. These techniques are light-scattering photometry, membrane osmometrv. .. and ultracentrifueation. The ultracentrifueation technique is typically covered, 0 t h in too bwat detail, in most physicd chemistry texts and will nut be dealt wirh here.

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Light-Scattering Photometry Raleigh showed in 1871 that induced oscillatory dipoles were developed when light passed through gases and that the amount of scattered light ( 7 ) was inversely proportional to the fourth power of the wavelength of light. This investigation was extended to liquids by Einstein and Smoluchowski in 1908. These oscillations reradiate the light energy to produce turbidity, or the Tyndall effect. Other sources of energy, such as X-rays or laser beams, may he used in place of light waves. For light-sratteringm&surenlents the total amount of the scattered light is deduced from the decrense in intensitv 01' the incident beam, 10,as i t passes through a polymer sample of length 1. This can he described in terms of Beer's law for the absorption of light as follows

I = e-il

Fiwe 1. Schamatic represenlatian of face extensim wwe fa(a) a rubber-like substance. (b) a simple solid (3).

reduces to an equation of a straight line,

I0

where T is the measure of the decrease of the incident-beam inteusitv per unit leneth of a eiven solution and is called the turhidiiy.The intensity of scattered light or turbidity (7)is proportional to the square of the difference between the index of refraction n of the polvmer solution and the solvent. to the weight-average moiechar weight of the polymer (MJ, and to the inverse fourth power of the wavelength of lieht used (A). Thus,

where the expression for the constant H is as follows

where no = index of refraction of the solvent, n = index of refraction of the solution, c = concentration, the virial constants B, C, etc., are related to the interaction of polymer with solvent, P(0) is the particle scattering factor, and N A is AVOgadro's number. The expression dnldc is the specificrefractive increment and is determined by taking the slope of the refractive index readings as a function of polymer concentration. In the determination of the weight-average molecular weight of polymer molecules in dust-free solutions, one measures the intensity of scattered light from a mercury arc lamp or laser a t different concentrations and at different angles (81, typically 0,90,45, and 135". The incident light sends out a scattering envelope which has four equivalent quadrants. The ratio of scattering a t 45' compared with that for 135' is called the dissymmetry factor or dissymmetry ratio Z. The reduced dissymmetry factor Zo is the intercept of the plot of Z as a function of concentration extrapolated to zero concentration. For polvmer solutions containing polymers of moderate to low iofecular weight, Zo is 1and eqn. ( I ) reduces to

At low concentrations of polymer in solution, the above 1032


When the ratio of the concentration r to the rurhidity r (related to the intensity of scarterinpc at Oand YOo) multiplied hv the constant H is plotted againstconcentrat@n, the interce*t of the extrapolated curve is the reciprocal of M, and the slope is thevirial constant B. Zois directly related to P(0), and hoth are related to hoth the size and shape of the scattering particle. As the size of the polymer chain approaches ahout onetwentieth the wavelength of incident light, scattering interference occurs giving a sratwring envelupe that is nolonger symmetrical. Here. the scattering dependency on molecular weight reverts hack to the virial relationship, thus a plot of H c / r versus C extrapnlated to zero poly_mer conrentration gives. as rhc intercewt. 1/%,P(8). nor 1,M,. The molecular weight for such situations i s found utilizing low-angle measurements or one of two treatments of the data-the Zimm or Dissymetry methods. These two approaches are described in most introductorv ~ o l v m e texts. r The student shoild hk aware of all these techniques for determining- ~. o l- v m eaverage r chain leneth. - . the association of each technique with a givenmolecular weight (i.e., weight or number average value) and the fundamental aporoaches and equations (exrluding memory of a d e ~ c r i p t i o ~"dHf " ) . Further, it is the opiniun of therommittee that most textsco into too great a detail regarding the ultracentrifugation tecGnique and that a less detailed approach is appropriate. Finally, it must be noted that ultracentrifugation can give several molecular weight values depending on the treatment of data and experimental design. Typically the apparatus is operated in such a mode that only the molecular weight of a monodisperse sample (as a specified monodisperse hiopolymer such as an enzyme) can he obtained. Additional measurements must he made along with specified modes of data treatment employed to arrive at molecular weights for heterogeneous samples. Rubber Elasticity In the thermodynamic analysis given previously we found that for a rubber-like substance (a disordered polymer system) the major contribution to. the retractive-fake comes

from the entropy decrease accompanying the deformation. This conclusion is reached solely on the basis of the Second Law and the experimental thermoelastic coefficient. This basis for elastic deformation is quite different as compared to other substances and accounts for the uniaue lone-ranee elasticity of rubber. For low-molecular-weig'ht s u b k n & the maior contribution to the retractive force has an enereetic basis. A typical example of a force-extension curve for a rubber-like substance. as comvared to low-molecular-weieht substance is given in'FigureAl.Rubber can sustain a very large deformation of the order of six- to tenfold and returns to its original state when the external force is removed. On the other hand, usual low-molecular-weight substances can only he extended very slightly and require a very large force, for this small extension. Inreality, it must be recognized that there could be, and in fact are, both entropic and enthalpic contributions to the retractive force. The formal thermodynamic analysis makes the point that there must be differences in the mdecular basis account for the elastic properties. The question then arises as to whether one can furnish a molecular understanding of ruhber elasticity. These elastic properties can be put on a quantitative molecular basis, and their understanding represents a classical achievement of physical science (7a, 8).As a first step we consider only entropic contributions to the retractive force. In terms of statistical mechanics, we seek the change in the number of configurations of the system upon deformation. In the undeformed state, a collection of chain molecules (crosslinked to prevent irreversible flow) has endto-end vectors that are distributed according to eqn. (1). When deformed, as for example in simple extension, components of the vectors in the direction of the deformation will he increased. Hence the more extended configurations will he favored. Consequently the number of configurations will he reduced. Consequently the number of configurations available to the svstem will decrease with an accom~anving . . " decrease in the entropy. The retractive force arises from the increase in entrovv on return to the undeformed state. This is the most state, the one having the maximum number of configurations and the maximum entropy. There is, of course, a complete analogy here to the calculation of the entropy decrease when the volume of an ideal gas is decreased (7a). The change in the number of configurations of the polymer chain can be calculated from eqn. (I),with the assumption of an affine deformation. This is one where the change in the coordinates of the chain vectors is proportional to the macroscopic deformation. I t is found that in simple extension the stress t is related to extension ratio a (final length over initial length) by the relation (7a, b )

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Here u is a number of chains and V the volume. This idealized equation of state for a rubber-like substance, taking into account only changes in configuration, is in very good agreement with exoeriments. We note that the stress is oro~ortionalto temperitwe a t constant length; for an ideal gas ihe pressure is proportional to the temperature at constant volume. Besides explaining elastic properties, and demonstrating quite forcibly that there can he more than just energetic contributions to the development of elasticity, the theory outlined represents a major achievement in explaining macroscopic mechanical properties in terms of microscopic molecular structure. The modulus of an elastomer increases with increasing temperature, just like the pressure of a gas increases with temperature. The equations relating Young's modulus to molecular properties can be written:

Chemical Kinetics Polymerization Kinetics Studies of polymerization kinetics develop some important general principles and yield interesting examples of chemical kinetics. They also lead to some basic information with regard to catalytic action. There are two general types of polymerization reactions. These have been named stepwise and chain polymerization, respectively. Each of these classes demonstrates very different principles of reaction kinetics. Polymer such as the polyamides (nylons), polyesters, and polyurethanes are formed by stepwise polymerization. Stepwise ~olvmerization involves a series of stenwise reactions in . which there is a slow buildup in the size of the molecules. Each step of most such reactions involves the elimination of a small molecule, as for example water. A simple example of the condensation process involves the reaction of an alcohol and an acid to form an ester. As ethyl alcohol and acetic acid react to form ethyl acetate water is eliminated, and the reaction stops. If, on the other hand, bifunctional reactants were used, such as a diacid and adialcohol then after the first step in the reaction the resulting species would still be functional. The different functional groups at each end of the molecule could then react either with themselves, or with the other bifunctional species present. Thus, for example, in the reaction of a diamine with a diacid to form a polyamide,

Each step of the reaction involves the elimination of a water molecule. The polymerization proceeds by this series of stepwise reactions. In order to develop high molecular weight species the reaction must be driven to a point where almost all the functional groups have reacted. I t is also possible to carry out this kind of polymerization with only one reacting species provided that both functional groups are on the same molecule, as for example o-hydroxy acids. Although the chemistry of the reaction illustrated is very straightforward there are some very interesting consequences with respect to the mechanisms and the resultine It has m d e n ~ l a weight r and moitcular wight disrrih~~tions. I)em demonstrated that the pr~ncipleof euual reactivitv 01' a11 functimal g r a ~ ~ p i a p p l tosrrpwisr ic.~ pulymeri7atiun I 7cj. 'l'he inherent reactivity of a lunct~onaly n u p 1s shown to be indeoendent of the molecular weieht. BV invokin~ u the .orinciplethat the reaction rate is independent of the molecular weieht of the soecies. it is oossible to derive the molecular weight distributions, and average molecular weight, as a function of the extent of the reaction (9).I t is found theoretically, and demonstrated experimen~ullythat, a most pruilnhle d i s t r ~ l ~ u t iri:sults m from a condensation reaction. Yoit stepwise prdymeri~atiunroipolyesters and polyamides yield molrr~liarweights which are on the order 01'al)our 20-10 X 10",involving 200 to 400 units. In the case of polyamides i t is possible to demonstrate very rapid polymerization by taking advantage of the reaction of a diamine with a di(acid chloride) a t the interface between two immiscible liquids. Therefore, the stepwise polymerization proceeds rapidly at room Volume 62


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temperature to produce a high molecular weight polymer without the necessitv of a catalvst - (10-12). . Another interesting ramification of the stepwise reaction is the ability to form branched molecules and network structures. If one of the original monomers contains more than two functional groups per molecule then a branching reaction is possible, and network-like structures can also be obtained instead of linear chains. In contrast to stepwise polymerization, chain polymerization is characterized by the fact that two molecules are joined into one without the loss of any small fragment, i.e., lowmolecular-weight species. Addition polymerization is representative of chain reactions as a class. The synthesis of longchain molecules bv a chain reaction is a verv-exnlicit . demonstration of this pr&ess since a very definite product is formed whose molecular weight can be predicted by calculation and actually confirmed by measurement (13). The most common organic compounds that participate in addition reactions are unsaturated species containing a carbon-carbon double bond. Because of the electronic nature of the double bond such species are effectively bifunctional and have the abilitv to form lone chains. Vinvl monomers and dienes are common examples of species which participate in addition ~olvmerization. In chain pblymerization the hasic mechanism involves the activation of the first monomer which then successivelv attacks and adds on to hundreds or thousands of additional monomers. This process results in the rapid growth of the individual polymer chain. The reaction can be broken down into three basic steps which are described in the following schematic manner. Initiation: M

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M*

(14)

Hence M represents the monomer unit and M* the activated species. In an unsaturated monomer, the initiation process involves the method whereby the double bond is attacked. This is eoverned bv the tvoe of catalvst or initiator that is used. T K ~initiationcan he classified aEcording to the type of active snecies that are ~roduced.There are three main varieties, a free radical, a cation, or an anion. Detailed analysis indicates that each of these species can play a very unique role in the polymerization. The active species can attack the double bonds, leading to the propagation step which is schematically illustraLed as follows:

r r in the growing chain where n is thr number of m ~ , n ~ n nunits hl,'. The process 1s terminated by coml~inatiunur disprw portionation.

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Termination by disproportionation: MN* M N Termination by combination: MN* MzN The basic scheme that has been outlined assumes that there are no side. or com~licatine.reactions. A common tvne -. of side reaction is chain transf;;, wherein the growth of a chain is terminated and the radical is transferred to unreacted monomer with a special chain transfer molecule added, such as dodecyl mercaptan, or to the medium in which the reaction is taking place. Reeardless of the details of the mechanisms. addition pulv~~erizintirm has the special characteristic invol'vin):rapid growth of each individual chain until it is terminat~.d.This proress clearly differs from rundensatiun polyrnerizi~tion where all the molt.cules rumbine wirh each othrr rlurinr: the course of the reaction and the average size graduall; increases. Addition polymerization has the interesting property that it is possihle to calculate the average chain size from the ki1034


netic scheme. In general the number average degree of polymerization Rate of propagation X" = (16) Rate of termination and transfer This eauation is a very eeneral one, which allows for all t w e s of termination and transfer step. By measuring x, the spe&s of the reaction can be probed. A more detailed approach suitable for use in introductory courses is contained-& most introductory polymer texts. Catalysis Students should be familiar with a range of topics associated with catalvsis includine the basis of the mechanisms and kinetics of enzyme action, synthesis of bothsmall and polymer compounds through stepwise, chain, etc., pathways, and fundamentals of gas-phase, solution, and solid catalysis. 11lustrations of solution and solid catalysis involving polymers abound and include detailed mechanistic and kinetic presentations, with most introductory polymer texts containing. .. ample examples. Topics that should be utilized include polymerization with complex coordination catalysts and the relation of such catalyst systems with the stereoregularity of the resulting A student should also be able to identifv isotactic. nolvmers. " syndiotactic, and atactic forms of vinyl polymers. The enzymes are polymers (macromolecules), and they are biological catalysts. Their action shows some resemblance t o the catalvtic action of small acids and bases but is considerablv more complicated. The details of the mechanisms of enzyme action are still beine" worked out. and much research remains to he done. One way in which enzymes differ from classical, small acids and bases is that they show amarked degree of catalytic specificity. Some enzymes act upon only one substrate and are said to show absolute specificity. A lower degree of specificity is shown by the proteolytic enzymes, which catalyze the hydrolysis of the peptide linkage, provided that certain chemical groups are present in the neighborhood of the linkaee. This is known as erouD snecificitv. Manv enzymes exKibit stereochemical s&~&&, in that they eatalvze the reactions of one stereochemical form and not the uther. The protroly~icrnzymes, for example, ratalyze only the hydrolssis of peptides made up from amino acids in the I. configurations (f4): All enzymes are basically proteins, hut they may be associated with nonprotein substances, known as coenzymes or prosthetic groups, which are essential to the action of the enzyme. Some enzymes are catalytically inactive in the ahsence of certain metal ions. For a number of enzymes the catalytic activity is due to a relatively small region of the protein molecule, referred to as the active center. In studying the kinetics of an enzyrne-catalyzed reaction the most reliable procedure is first to make rate measurements in the early stages of the reaction. By following the change in concentration of substrate or a product, the initial slope of a coucentration-time curve leads at once to the initial rate of the reaction. The variation of this initial rate with substrate concentration can then be investigated. This is a much more reliable procedure than to allow the reaction to proceed to a ereater extent and to analvze the concentration-time curves 6y the method of integracon. The difficulty with this latter orocedure is that nroducts freauentlv have an imnortant effect on the rates of enzyme reactions. The concentration-time curves will be influenced hv such effects and sometimes will be misleading. The s~ecifickinetic a~nroachesare described in detail in must in;rodurtory biuche~istrstexts and in ref. (/.In).It is rra~mrnrnrlnlthat the hliohaelis-Menten ruuation br derived and discussed and its relationship to the ~ a i ~ m uadsorption ir isotherm be noted.

.

Idealized ModulusTemperature Behavior

where the modulus is below lo5Pascals. The viscous region will be discussed further below. The rate of flow is related to slow disentanglement and slippage of polymer chains past each other, and follows the Williams, Landel, and Ferry (WLF) equation in which the constants B and C are 17.44 and 51.6, and T, is the glass transition temperature (17) -B(T - T,) l0g(a=c+(T-Tg) The polymer molecular weight g is related to the intrinsic viscosity, [?I by the parameters K and a as follows.

Temperature Figure 2. Modulus-temperaturebehavior faan amorphous polymer, illushating both (a)crosslinked and (b) linear viscoalastic behavior (Is). Vlscoelastic Behavlor Rheology The branch of scienw related to the study of dt.formation and flow of macerials was given the name rheolorv with the prefix rhea derived f r o m t h e Greek term rheas; meaning current or flow. The study of rheology includes two vastly different branches of mechanics called fluid and solid mechanics. The polymer chemist is usually concerned with viscoelastic materials that act as both solids and fluids (15). The elastic component is dominant in solids, hence their mechanical orooerties mav be described bv Hooke's law. which statesiha; the applieh stress t.7) is proportional to thd resultant strain (? I b ~ is n indeoendent of the rate of this strain

Viscometry does not lead to absolute molecular weight val'ues but rather is only a relative measure of a polymer's molecular weiaht. Thus, viscometry measurements must be correlated with an "ahsolute molecular weight method" such as light-svattering photometry. Experimentally 1h6 visroslts is determined for several polymer samples varying in only molecular weight. Then the molecular weight of each sample is determined using is conan absolute method. A plot of log [TI versus log structed enabling the determination of a and K. After c a l c u l a t i o ~ an f a and a K value for a given polymer-solvent pair, M can be easily calculated using a determined [TI] and the previously described relationship. A simple introduction to the viscosity of dilute polymer solutions should be made (19,20). Llterature Cited (1) J. CWEM. EDUC..62.780 (19851. (2) Mandelkern, L., "Lnhoduetian to Macromoledes,"

2nd ed.. Sprinper-Verlap. 1983, p. 38. (31 Flory, P. J.,"Statistical Mechanics of Chain Molceulea," Interscience, 1969. (4) Poland. D.. and Scharaea. H. A.."Thwrv of Helix-Coil Trsnsitians in Biaoolmem!' ... . . ...-., ..... (5) Mattice, W. L., J.CHEM. EDUC.,58.911 (1981). 161 Treloar, L. R. G., "The Physics of Rubber Elaatieity," Clamnder Press, Oxford.

."*". ,a"a

(7) Flory, P. J., "Principles of Polymer Chemistry," Cornell U. Press, 1958: (a) Chapter Xi, p. 432: lb! p. 77ff:(d p. 318ff. (8) Florv. P. J.. and Vrii. A,. J. Amer Cham. Soc. 85.3548 119631.

.".-,."...

1121 Cnrraher. C..an d Prealon, J.. (Editors),"Interfacial Synthesk," Val. 3. Dekker, New

Stress is equal to force per unit area, and strain or elongation is the extension per unit length. Theviscous component is dominant in liquids, hence their flow properties may be described by Newton's law, which states that the applied stress S is proportional to the rate of strain dyldt but is independent of the strain y or applied velocity gradient. The symbol? is sometimes used for strain rate.

Both Hooke's and Newton's laws are valid for small changes and both are useful in studying the effect of stress on viscoelastic materials. The modulus, or stiffness, of a polymer decreases through several orders of magnitude as the temperature increases (16-18 (see Fiz. - 2). . Five reeions are usuallv identified (16): . . (1)The glassy region, whereUonlybond bending on stretching takes vlaces:. (2) . . the alass transition reeion. - . where increasine energy permits the onset of long-range coordinated molecular motion in the polymer chain; (3) the rubbery plateau, which describes the behavior of network systems (a), but for a short temperature interval alio describes linear polynlers (b! if the molecular weight is sufficiently high. Region (4) is called the rulhery flow regiun, whrre ior linear pdymers. both elastic and viscous features combine. ~ a s t l y , t h highe est temperatures are in region ( 5 ) , the viscous flow region,

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."...,.

(13) Morton, M.,J. CHEM. EDUC., 50,740 (1973). (14) Laidler, K. J., '"Physical Chemiahy with Biological Applications."Benjamine, Menla Park, CA, 1978: 1s) pp. 427.458: (b! 46%505. (15) Seymour. 8.. and Carraher, C.,"PolmerChemistry: An Intmduetion,"Dekker, New York, 1981,pp.23-40. (161 Tobolnky, A. V.. "Properties and Structure of Polymers: John Wiley, New York. 1960.

(171 Williams, M. L., Landel.

a. F., and

Ferry, J. D.. J . ~

m Chem. m sor., 77. 3701

Recent General References Hcar1e.J W S.. 13015mrmandTur#rPropcn.rr" Elln H ~ ~ n o o d . ~ ' l ~ r h cIn' Kc.r19h2 . R n l r ~ w r ?F . 'PI.III I I . L ~PI,.snwr ~ >\.wm."2nrl 1.1 . \ I r G r o * - l l l l Nr r Ydrk 19rl

H. H . s n d ~'arrahcr.C E . l r . Polymer I hrm..rr\ 4n i n r r d u r i ~ o n ' \larcel Dokker. New York, 1981. Allmek, H. R., and Lsmpe, F. W., "Contemporary Polymer Chemistry," Prentiee~Hall, Endewoad Cliffs. NJ. 1981. young, k. J.. "lnfrod&tionto Polymers," Chapman and Hell, London, 1981. Wall, F. T.,"ChemicslThermdvnamics," W. H. Freeman &Co., 1965,Ch.15 Billmeyer, F. W. J r . , " ~ e r t b o o k d~o1ym.r Seicnee," 3rd ed., Wiley, New York, 1984. Mandelkern, L., "An lntroductian to Maaomolodos,"2nd ed., Springer-Verlsg. New York, -ir\rnour

1983.

Eliss. H. G., '"Macromolmles:

Vals. 1and 2, Plenum, New York. 1984.

Appendix: Polymer-Oriented Physical Chemistry Questions 1. Globular polymers (a) have an impact on intrinsic viscosity that is independent of molecular weight. (b) can be expected to show more flow birefringence than a random coil polymer. Volume 62


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(c) are more likely to exist in an ideal solvent than random coil polymers. (d) exhibit large values of the Mark-Houwink parameter "a". 2. A stretched rubber band (a) is more nearly syndiotaetie than an unstretched rubber band. (b) will heat on release of tension. (c) will increase its tension when warmed. (d) has a significantly smaller volume than an unstretched rubber band. 3.

A polypeptide that can exist in helical form (a) is more likely to exist in the helical form in a bydrogen-bonding

solvent. (h) is more likely to exist in the random-coil configuration in the presence of electrolytes. rc) will have mow rffrrt on visccm~tyin rhe h e l i ~ d form than in the random coil contiyuration. (d) will always effect an entropy increase on uncoiling.

4. A weight-average molecular weight (a) is smaller than the numher-average molecular weight for polydisperse samples only. (b) can be obtained from osmotic pressure measurements. (c) can be obtained from light-scattering measurements only if the parameters of the Mark-Houwink equation are known from viscosity measurements. (d) can he obtained from sedimentation experiments. 5. A multiole-viscositvmotor oil such as 10W-40has a viscositv that changes more slowly with temperature than regular motor oil. The multiple viscosity motor oil. (a) contains a silane additive that reduces the activation energy for viscous flow. (b) polymerizes a t high temperatures to increase the viscosity. (c) contains a polymer that expands its coil dimensions with increasing temperature through such mechanisms as increased solubility, increased second virial coefficient, andlor a shift in earbon-carbon bond population from trans to gauche canformations. (d) consists only of unbranched hydrocarbons with an even number of carbon atoms.

1036


6. Which of the following is a true statement concerning distribution functions? (a) The weight distrihution function for a polydisperse polymer, F d M ) , can always be obtained from the corresponding number distribution function.. FdM). ... . . (b) The Planek radiation law expressed as a function of wavelength and the law expressed as a function of frequency have maxima at points corresponding to the same energy. (c) The electron density distribution for a Is orbital has a maximum at a distance from the origin equal to the radius of the first Bohr orbit. (d) The distribution of molecular speeds in two dimensions has a maximum at the origin. 7. The osmotic pressure rise of an aqueous solution containing 2.00 g of polyvinyl alcohol per liter was found to he 4.68 mm above that of the pure solvent a t 20°C. The densities of water and mercury are 1.00 and 13.6 glcm3. The molecular weight of the polymer in Daltons is approximately (a) 7.8X 103 (b) 1.1X 105 ( e ) 1.4 X 105 (d) none of the above

8. The equation expressing the melting temperature, T,, as a function of its nnmber-average molecular weight, M., can be written:

1

T,

1

R 2Mo

lam AH, M,

where Mo represents the monomer molecular weight, AH, the heat of melting, P, the melting temperature of an infinite moleeeas constant. Which of ular weieht linear oolvmer., and R is the " " the following is true?

..

(a) The constant 2 multiplying the mer molecular weight appears

because of entropic effects. (h) The constant 2 appears because alinear polymer has two ends, each of which constitutes an impurity in an otherwise repetitious structure. (c) The melting temperature of polymers is invariant with molecular weight, so the above equation is untrue. (d) Since this equation expresses the melting point as a function of molecular weight, the heat of polymerization should replace the heat of melting.

Polymer principles in the undergraduate physical chemistry course

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