Fundamentals of Polymer Engineering Part 11 ppt

It is surprising that in one of the earliest and most successful models, called thefreely jointed chain [2,3], we can entirely disregard the chemical nature of thepolymer and treat it as

or so, rubber can be stretched to as much as 10 times its length without damage.Furthermore, the stress needed to achieve this deformation is relatively low Thus,polymers above Tg are soft elastic solids; this property is known as rubberlikeelasticity Other extraordinary properties of rubber have also been known for along time Gough’s experiments in the early 1800s revealed that, unlike metals, astrip of rubber heats up on sudden elongation and cools on sudden contraction[1] Also, its modulus increases with increasing temperature These properties arelost, however, if experiments are performed in cold water Explaining theseremarkable observations is useful not only for satisfying intellectual curiosity butalso for the purpose of generating an understanding that is beneficial for tailoring

the properties of rubberlike materials (called elastomers) for specific applications.Recall that rubber (whether natural or synthetic) is used to manufacture tires,adhesives, and footwear, among other products Note also that because polymerproperties change so drastically around Tg, the use temperature of most polymers

is either significantly below Tg(as in the case of plastics employed for structuralapplications) or significantly above Tg (as in the case of elastomers)

Chemically, rubber is cis-1,4-polyisoprene, a linear polymer, having amolecular weight of a few tens of thousands to almost four million, and a widemolecular-weight distribution The material collected from the rubber tree is alatex containing 30–40% of submicron rubber particles suspended in an aqueousprotein solution, and the rubber is separated by coagulation caused by theaddition of acid At room temperature, natural rubber is really an extremelyviscous liquid because it has a Tg of
70C and a crystalline melting point ofabout
5C It is the presence of polymer chain entanglements that prevents flowover short time scales

In order to explain the observations made with natural rubber and otherelastomers, it is necessary to understand the behavior of polymers at themicroscopic level This leads to a model that predicts the macroscopic behavior

It is surprising that in one of the earliest and most successful models, called thefreely jointed chain [2,3], we can entirely disregard the chemical nature of thepolymer and treat it as a long slender thread beset by Brownian motion forces.This simple picture of polymer molecules is developed and embellished in thesections that follow Models can explain not only the basics of rubber elasticitybut also the qualitative rheological behavior of polymers in dilute solution and asmelts The treatment herein is kept as simple as possible More details areavailable in the literature [1–7]

FREELY JOINTED CHAIN

One of the simplest ways of representing an isolated polymer molecule is bymeans of a freely jointed chain having n links each of length l Even though realpolymers have fixed bond angles, such is not the case with the idealized chain Inaddition, there is no correspondence between bond lengths and the dimensions ofthe chain The freely jointed chain, therefore, is a purely hypothetical entity Itsbehavior, however, is easy to understand In particular, as will be shown in thissection, it is possible to use simple statistical arguments to calculate theprobability of finding one end of the chain at a specified distance from theother end when one end is held fixed but the other end is free to move at random.This probability distribution can be coupled with statistical thermodynamics toobtain the chain entropy as a function of the chain end-to-end distance The

expression for the entropy can, in turn, be used to derive the force needed to holdthe chain ends a particular distance apart This yields the force-versus-displace-ment relation for the model chain If all of the molecules in a block of rubber actsimilarly to each other and each acts like a freely jointed chain, the stress–strainbehavior of the rubber can be obtained by adding together contributions fromeach of the chains Because real polymer molecules are not freely jointed chains,the final results cannot be expected to be quantitatively correct The best that wecan hope for is that the form of the equation is correct This equation obviouslyinvolves the chain parameters n and l, which are unknown If we are lucky, all ofthe unknown quantities will be grouped as one or two constants whose values can

be determined by experiment This, then, is our working hypothesis

To proceed along this path, let us conduct a thought experiment Imagineholding one end of the chain fixed at the origin of a rectangular Cartesiancoordinate system (as shown in Fig 10.1) and observe the motion of the otherend You will find that the distance r between the two ends ranges all the wayfrom zero to nl even though some values of the end-to-end distance occur morefrequently than others In addition, if we use spherical coordinates to describe thelocation of the free end, different values of y and f arise with equal frequency As

a consequence, the magnitude of the projection on any of the three axes x, y, and z

of a link taken at random will be the same and equal to l=pffiffiffi3

To determine the probability distribution function for the chain end-to-enddistance, we first consider a freely jointed, one-dimensional chain having links oflength lx¼ l=pffiffiffi3

, which are all constrained to lie along the x axis What is theprobability that the end-to-end distance of this one-dimensional chain is mlx? The

FIGURE10.1 The unconstrained freely jointed chain

answer to this question can be obtained by analyzing the random walk of a personwho starts out from the origin and takes n steps along the x axis; nþof these stepsare in the positive x direction and n
are in the negative x direction, and there is

no relation between one step and the next one Clearly, m equals ðnþ
n
Þ.From elementary probability theory, the probability that an event will occur

is the ratio of the number of possible ways in which that event can occur to thetotal number of events As a consequence, the probability, p(m), of obtaining anend-to-end distance of mlx is the number of ways in which one can take nþforward steps and n
backward steps out of n steps divided by the total number ofways of taking n steps The numerator, then, is the same as the number of ways ofputting n objects (of which nþare of one kind and n
are of another kind) into acontainer having n compartments This is n!=ðnþ!n
!Þ Because any given stepcan either be a forward step or a backward step, each step can be taken in twoways Corresponding to each way of taking a step, the next step can again betaken in two ways Thus, the total number of ways of taking n steps is 2n, whichgives us

r

1 þmn

 ðnþmþ1Þ=2

1
mn

 ðn
mþ1Þ=2
1

ð10:2:5ÞTaking the natural logarithm of both sides of Eq (10.2.5) and recognizing that

 

m22nþm2

Neglecting the very last term in Eq (10.2.7),

pðxÞ dx ¼ ð2npl2xÞ
1=2e
x 2 =2nl 2

In order to extend the one-dimensional results embodied in Eq (10.2.9) tothe three-dimensional case of practical interest, we use the law of joint prob-ability According to this law, the probability of a number of events happeningsimultaneously is the product of the probabilities of each of the events occurringindividually Thus, the probability, pðrÞ dr, that the unconstrained end of thefreely jointed chain lies in a rectangular parallelepiped defined byx; y; z; x þ dx; y þ dy, and z þ dz (see Fig 10.1) is the productpðxÞ dx pð yÞ dy pðzÞ dz, where pð yÞ dy and pðzÞ dz are defined in a manneranalogous to pðxÞ dx Therefore,

!

dx dy dzð10:2:10ÞDenoting the sum ðx2þ y2þ z2Þ as r2 and recalling that l2¼ l2¼ l2

z ¼ l2=3,pðrÞ dr ¼ 3

again Eq (10.2.11), but with the right-hand side modified by replacing dx dy dzwith 4pr2dr, the volume of the spherical shell Finally, then, we have

pðrÞ dr ¼ 3

2npl2

 3 =2

e
3r2=2nl24pr2 dr ð10:2:12Þwhich represents the probability that the free end of the chain is located at adistance r from the origin and contained in a spherical shell of thickness dr This

is shown graphically in Figure 10.2 Note that the presence of r2 in Eq (10.2.12)causes pðrÞ to be zero at the origin, whereas the negative exponential drives pðrÞ

to zero at large values of r As seen in Figure 10.2, pðrÞ is maximum at anintermediate value of r2 Also, because the sum of all the probabilities must equalunity,Ð1

0 pðrÞ dr ¼ 1

At this point, it is useful to make the transition from the behavior of a singlechain to that of a large collection of identical chains It is logical to expect that theend-to-end distances traced out by a single chain as a function of time would bethe same as the various end-to-end distances assumed by the collection of chains

at a single time instant Thus, time averages for the isolated chain ought to equalensemble averages for the collection of chains Using Eq (10.2.12), then, theaverage values of the chain’s end-to-end distance and square of the chain’s end-to-end distance are as follows:

hri ¼

ð1 0

where the angular brackets denote ensemble averages Because the fully extendedlength of the chain (also called the contour length) is nl, Eq (10.2.14)demonstrates that the mean square end-to-end distance is very considerablyless than the square of the chain length Therefore, the freely jointed chainbehaves like a random coil and this explains the enormous extensibility of rubbermolecules.

Having obtained the average value of the square of the chain end-to-enddistance and the distribution of end-to-end values about this mean, it is worthpausing and again asking if there is any relation between these results and resultsfor real polymer molecules In other words, how closely do freely jointed chainsapproximate actual macromolecules? If the answer is ‘‘not very closely,’’ then how

do we modify the freely jointed chain results to make them apply to polymers?The first response is that most polymer molecules do, indeed, resemble longflexible strings This is because linear (unbranched) polymers with a large degree

of polymerization have aspect ratios that may be as high as 104 They are thusfairly elongated molecules Furthermore, despite the restriction to fixed bondangles and bond lengths, the possibility of rotation about chemical bonds meansthat there is little correlation between the position of one bond and another onethat is five or six bond lengths removed However, two consequences of theserestrictions are that the contour length becomes less than the product of the bondlength and the number of bonds and that the mean square end-to-end distancebecomes larger than that previously calculated

If bond angles are restricted to a fixed value y, the following can be shown[4]:

hr2i ¼ nl2ð1
cos yÞ

If, in addition, there is hindered rotation about the backbone due to, say, stericeffects, then we have

hr2i ¼ nl2ð1
cos yÞð1 þ coshfiÞ

ð1 þ cos yÞð1
coshfiÞ ð10:2:16Þwhere hf2i is the average value of the torsion angle Small-angle neutronscattering data have supported this predicted proportionality between hr2i and nl2.Because hr2i increases with each additional restriction but remains propor-tional to hr2i for a freely jointed chain, we can consider a polymer molecule afreely jointed chain having n0links, where n0is less than the number of bonds, butthe length of each link l0is greater than the bond length, so that hr2i is again n0l02and the contour length is n0l0

Example 10.1: Polyethylene has the planar zigzag structure shown in Figure10.3 If the bond length is l and the valence angle y is 109:5, what are the

contour length R and the mean square end-to-end distance? Let the chain have nbonds and let there be free rotation about the bonds.

Solution: From Figure 10.3, it is clear that the projected length of each link is

l sinðy=2Þ Using the given value of y and noting that there are n links, the fullyextended chain length is given by

R ¼ nl sinð54:75Þ ¼

ffiffiffiffi23

rnl

The mean square end-to-end distance is obtained from Eq (10.2.15) as follows:

hr2i ¼ 2nl2

When the mean square end-to-end distance of a polymer is given by Eq.(10.2.16), the polymer is said to be in its ‘‘unperturbed’’ state What causes thepolymer to be ‘‘perturbed’’ is the fact that in the derivation of Eq (10.2.16), wehave allowed for the possibility of widely separated atoms that make up differentportions of the same polymer molecule to occupy the same space In reality, thosearrangements that result in overlap of atoms are excluded This is known as theexcluded-volume effect, and it results in dimensions of real polymer moleculesbecoming larger than the unperturbed value It is customary to quantify thisphenomenon by defining a coil expansion factor that is the ratio of the root meansquare end-to-end distance of the real chain to the corresponding quantity for theunperturbed chain In a very good solvent, there is a further increase in size, asdetermined by intrinsic viscosity measurements, and the coil-expansion factor canbecome as large as 2 In a poor solvent, on the other hand, the molecule shrinks,and if the solvent quality is poor enough, the coil expansion factor can becomeunity In such a case, the solvent is called a theta solvent, and we have the thetacondition encountered earlier in Chapter 9 It is, therefore, seen that the thetacondition can be reached either by changing temperature without changing thesolvent or by changing the solvent under isothermal conditions

In closing this section, we re-emphasize that the size of a polymer moleculemeasured using the light-scattering technique discussed inChapter 8is the mean

FIGURE10.3 The planar zigzag structure of polyethylene

square radius of gyration hs2i For a freely jointed chain this quantity, defined asthe square distance of a chain element from the center of gravity, is given by

hs2i ¼1

The radius of gyration is especially useful in characterizing branched moleculeshaving multiple ends where the concept of a single end-to-end distance is notparticularly meaningful

If we return to the unconstrained chain depicted inFigure 10.1and measure thetime-dependent force needed to hold one of the chain ends at the origin of thecoordinate system, we find that the force varies in both magnitude and direction,but its time average is zero due to symmetry If, however, the other chain end isalso held fixed so that a specified value of the end-to-end distance is imposed onthe chain, the force between the chain ends will no longer average out to zero.Due to axial symmetry, though, the line of action of the force will coincide with r,the line joining the two ends For simplicity of analysis, let this line be the x axis

In order to determine the magnitude of the force between the chain ends, let

us still keep one end at the origin but apply an equal and opposite (external) force

f on the other end so that the distance between the two ends increases from x to

x þ dx The work done on the chain in this process is

where k is Boltzmann’s constant In the present case, pðxÞ is given by Eq (10.2.9)

f ¼3kT

which is a linear relationship between the force and the distance between chainends and is similar to the behavior of a linear spring The constant ofproportionality, 3kT=nl2, is the modulus of the material and its value increases

as temperature increases This explains why a stretched rubber band contracts onheating when it is above the polymer glass transition temperature

The positive force f in Eq (10.3.7) is externally applied and is balanced by

an inward-acting internal force, which, in the absence of the external force, tends

to make the end-to-end distance go to zero This, however, does not happen inpractice because the spring force is not the only one acting on the chain; theequilibrium end-to-end distance is given by a balance of all the forces acting onthe polymer molecule This aspect of the behavior of isolated polymer moleculeswill be covered in greater detail in the discussion of constitutive equations fordilute polymer solutions inChapter 14

If we were not aware of the assumptions that have gone into the derivation

of Eq (10.3.7), we might conclude that the force between the chain endsincreases linearly and without bound as x increases Actually, Eq (10.3.7) isvalid only for values of x that are small compared to the contour length of thechain For larger extensions exceeding one-third the contour length, f increasesnonlinearly with x, and we know that for values of x approaching nl, chemicalbonds begin to be stretched It can be shown that the right-hand side of Eq.(10.3.7) is merely the first term in a series expansion for f [1]:

f ¼kT

l

3x

nlþ95

xnl

 3

þ297175

xnl

 

ð10:3:8Þ

where L
1is called the inverse Langevin function The Langevin function itself isdefined as

A closer examination of Eq (10.3.3) reveals a significant differencebetween the nature of rubbers and the nature of crystalline solids In general, fincludes contributions due to changes in entropy as well as changes in internalenergy In crystalline solids, the change in entropy on deformation is small and all

FIGURE10.4 Force-extension relation for a freely jointed chain (Reprinted fromTreloar, L R G.: The Physics of Rubber Elasticity, 3rd ed., Clarendon, Oxford, UK,

1975, by permission of Oxford University Press.)

the work goes into increasing the internal (potential) energy For rubberypolymers, on the other hand, the entropy change dominates and f depends entirely

on changes in entropy It is for this reason that polymer molecules are said to act

as entropy springs Note that the spring constant decreases (i.e., the springbecomes softer) as the polymer chain length increases Also, because DU ondeformation is zero, a consequence of entropic elasticity is that the work done onstretching a rubber must result in a release of heat if the process is isothermal Ifthe stretching is rapid, however, adiabatic conditions may result so that thetemperature rises The reverse situation occurs when the stretched rubber isreleased For crystalline materials, on the other hand, stretching results in astorage of energy On removal of load, no work is done against any external forceand the recovered internal energy shows up as an increase in temperature

In this section, we are interested in determining how a block of rubber deformsunder the influence of an externally applied force The procedure for doing this isthe same as the one employed for the isolated chain in the previous section Weassume that there are N chains per unit volume, and each behaves like an isolatedchain in its unstrained, equilibrium state When the block of rubber deforms, eachchain making up the block of rubber deforms as well It is assumed that thedeformation is affine; that is, there is no slippage past chains and the macroscopicstrain equals the microscopic strain In other words, changes in the length ofindividual chains correspond exactly to changes in length of corresponding linesdrawn on the exterior of the bulk rubber

This assumption makes it possible to calculate the change in entropy ondeformation of a single chain for a specified macroscopic strain A summationover all chains gives the macroscopic change in entropy of the rubber block, andthe subsequent application of Eq (10.3.3) yields the desired force or stresscorresponding to the imposed strain Let us illustrate this process for someidealized situations The more general case will be considered later

Before proceeding further, we must define strain In a tensile test, we findthat materials such as metals extend only by 1% or less We, therefore, definestrain as the increase in length divided by either the original length or the finallength For rubbers, however, a doubling in length is easily accomplished, and theinitial length l0 and final length l are dramatically different Consequently, themeasure of infinitesimal strain that works for metals is inappropriate in this case;

a measure of finite strain is needed instead One popular measure is the Henckystrain ln l=l0 and another is the extension ratio l ¼ l=l0 The latter quantity ismore easily related to the force acting on one face of a block of rubber.Consider, for example, a normal force F acting perpendicular to one face of

an initially unstrained cube of rubber of edge l0 Under the influence of this force,

the cube transforms into a rectangular prism having dimensions l1; l2, and l3, asshown in Figure 10.5 If we define l1as l1=l0, l2as l2=l0, and l3as l3=l0, then theaffine deformation assumption implies that the coordinates of the end-to-endvector of a typical polymer chain change from ðx0; y0; z0Þ to ðl1x0; l2y0; l3z0Þ.Under this change in dimensions, the change in entropy of the chain is, fromEqs (10.2.11) and (10.3.4), as follows:

Because rubber is incompressible, its volume does not change on tion Therefore, it must be true that

From the definition of l1 it is obvious thatDl1 equalsDx=l0 Because W mustalso equal
Ð

F dx, F is obtained by dividing the right-hand side of Eq (10.4.6)

by l0 and differentiating the result with respect to l1 Thus,

F

l2¼ NkTðl1
l
21 Þ ð10:4:7Þ

It can be shown that the form of this equation is unaffected by the presence ofchains of unequal lengths; only the numerical value of the coefficient changes.The quantity NkT is called the modulus G of the rubber The left-hand side of

Eq (10.4.7) is recognized to be the stress based on the undeformed area.Example 10.3: When rubber is brought into contact with a good solvent, itswells in an isotropic manner Consider a cube of rubber, initially of unit volume,containing N polymer chains If in the swollen state the polymer volume fraction

is f2and the length of each edge is l, how much work is done in the process ofswelling?

Solution: Here, we use Eq (10.4.4), with each extension ratio being equal to l.Note that Eq (10.4.5) does not apply because there is an obvious increase involume The total volume of the swollen rubber is equal to 1=f2, so l ¼ f
1=32 Consequently,

W ¼3NkT

2 ðf
2=32
1ÞThis problem considers a particular kind of deformation–uniaxial extension Thesame procedure can be applied to other kinds of deformation, and the result is a

‘‘material function’’ or, in the case of rubber, a material constant that relates acomponent of the stress to a component of the strain imposed on the material.More generally, though, we can determine the relationship between an arbitrary,three-dimensional deformation and the resulting three-dimensional stress Such arelationship is called the stress constitutive equation We will develop such arelationship for rubbers after we review the definitions of stress and the strain inthree dimensions

If we isolate a rectangular parallelepiped of material having infinitesimal sions, as shown inFigure 10.6, we find that two kinds of forces act on the materialelement These are body forces and surface forces Body forces result from theaction of an external field such as gravity upon the entire mass of material Thus,the force of gravity in the z direction is gzr dx dy dz, where gz is the component

dimen-of the acceleration due to gravity in the positive z direction Surface forces, on theother hand, express the influence of material outside the parallelepiped butadjacent to a given surface Dividing the surface force by the area on which itacts yields the stress vector Because the parallelepiped has six surfaces, there aresix stress vectors Because each of the 6 vectors can be resolved into 3components parallel to each of the 3 coordinate axes, we have a total of 18components These are labeled Tij, where the two subscripts help to identify aspecific component The first subscript, i, identifies the surface on which the forceacts; the surface, in turn, is identified by the direction of the outward drawnnormal If the normal points in the positive coordinate direction, the surface is apositive surface; otherwise it is negative The second subscript, j, identifies thedirection in which the stress component acts According to convention, a stresscomponent is positive when directed in the positive coordinate direction on apositive face It is also positive when directed in the negative direction on anegative face Nine of the 18 components can be represented using a 3  3matrix, called the stress tensor:

Tzz, for example, is the z component of the stress vector acting on the face whoseoutward drawn normal points in the positive z direction; Tzyis the correspondingcomponent acting in the y direction These are shown in Figure 10.6 The othernine components are the same as these, but they act on opposite faces

By means of a moment balance on a cubic element, it can be shown (as inany elementary textbook of fluid mechanics) that Tijequals Tji Thus, only six ofthe nine components are independent components The utility of the stress tensor

is revealed by examining the equilibrium of the tetrahedron shown inFigure 10.7

FIGURE10.6 The stress matrix (tensor)