Polymer Solution

8/7/2019 Polymer Solution

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General and thermodynamic terms


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Polymer Dissolution

What is solubility and what it depends on?

It should be pointed out that not all polymers can be dissolved,

and even though when they can, the dissolution process may take

up to several days or weeks. According to Rosen (1982), there is

an assembly of general rules for polymer solubility, based on

experimental observations, from which interesting conclusions

can be obtained.

Thus, it is well known that the dissolution of polymers depends

not only on their physical properties, but also on their chemical

structure, such as: polarity, molecular weight, branching,

crosslinking degree, and crystallinity. The general principle that

states like dissolves like is also appropriate in the case ofpolymers. Thus, polar macromolecules like poly (acrylic acid),

poly (acrylamide) and polyvinyl alcohol, among others, are

soluble in water. Conversely, nonpolar polymers or polymer

showing a low polarity such as polystyrene, poly(methyl

methacrylate), poly(vinyl chloride), and poly(isobutylene), are

soluble in nonpolar solvents.

On the other hand, the molecular weight of polymers plays an

important role in their solubility. In a given solvent at a particulartemperature, as molecular weight increases, the solubility of a

polymer decreases. This same behavior is also noticed as

crosslinking degree increases, since strongly cross-linked

polymers will inhibit the interaction between polymer chains and

solvent molecules, preventing those polymer chains from being

transported into solution.

A similar situation occurs with crystalline macromolecules,

although in such a case the dissolution can be forced if an

appropriate solvent is available, or warming the polymer up to

temperatures slightly below its crystalline melting point (Tm). For

example, highly crystalline linear polyethylene (Tm = 135C) can

be dissolved in several solvents above 100C. Nylon 6.6 (Tm =

265C), a crystalline polymer which is more polar than

polyethylene, can be dissolved at room temperature in the


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presence of solvents with enough ability to interact with its

chains, through for example, hydrogen bonding. Branched

polymer chains generally increase solubility, although the rate, at

which this solubility occurs, depends on the particular type of

branching. Chains containing long branches, cause dense

entanglements making difficult the penetration of solvent

molecules. Therefore the rate of dissolution in these casesbecomes slower than if it was short branching, where the

interaction between chains is practically non-existent.

How a Polymer gets dissolved?

As said earlier, the dissolution of a polymer is generally a slow

process, which can take even several weeks, depending on the

structure and the molecular weight of a given polymer.

When a low molecular weight solute such as sucrose is added to

water, the dissolution process takes place immediately. The sugar

molecules leave the crystal lattice progressively, disperse in

water, and form a solution.

But polymer molecules are rather different. They constitute long

chains with a large number of segments, forming tightly folded

coils which are even entangled to each other. Numerous cohesive

and attractive both intra and intermolecular forces hold thesecoils together, such a dispersion, dipole-dipole interaction,

induction, and hydrogen bonding (Figure 1a).

Based on these features, one may expect noticeable differences

in the dissolution behavior shown by polymers. Due to their size,

coiled shape, and the attraction forces between them, polymer

molecules become dissolved quite slowly than low molecular

weight molecules. Billmeyer Jr. (1975) points out that there are

two stages involved in this process: in the first place, the polymer

swelling and next the dissolution step itself.

When a polymer is added to a given solvent, attraction as well as

dispersion forces begin acting between its segments, according to

their polarity, chemical characteristics, and solubility parameter.

If the polymer-solvent interactions are higher than the polymer-

polymer attraction forces, the chain segment start to absorb


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solvent molecules, increasing the volume of the polymer matrix,

and loosening out from their coiled shape (Figure 1b). We say the

segments are now "solvated" instead of "aggregated", as they

were in the solid state.

The whole "solvation-unfolding-swelling" process takes a long

time, and given it is influenced only by the polymer-solventinteractions, stirring plays no role in this case. However, it is

desirable to start with fine powdered material, in order to expose

more of their area for polymer-solvent interactions.

When crystalline, hydrogen bonded or highly cross-linked

substances are involved, where polymer-polymer interactions are

strong enough, the process does stop at this first stage, giving a

swollen gel as a result.

If on the contrary, the polymer-solvent interactions are still

strongly enough, the "solvation-unfolding-swelling" process will

continue until all segments are solvated. Thus, the whole loosen

coil will diffuse out of the swollen polymer, dispersing into a

solution. At this stage, the disintegration of the swollen mass can

be favored by stirring, which increases the rate of dissolution.

However, once all the chain segments have been dispersed in the

solvent phase, they still retain their coiled conformation, yet theyare now unfolded, fully solvated, and with solvent molecules

filling the empty space between the loosen segments. Hence, the

polymer coil, along with solvent molecules held within, adopts a

spherical or ellipsoid form, occupying a volume known as

hydrodynamic volume of the polymer coil (Figure 1c).

The particular behavior shown by polymer molecules, explains the

high viscosity of polymer solutions. Solvent and low molecular

weight solutes have comparable molecular size, and the solute

does not swell when dissolving. Since molecular mobility is not

restricted, and therefore intermolecular friction does not increase

drastically, the viscosity of the solvent and the solution are

similar. But the molecular size of polymer solutes is much bigger

than that of the solvent. In the dissolution process such molecules

swell appreciably, restricting their mobility, and consequently the


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intermolecular friction increases. The solution in these cases,

becomes highly viscous.

Figure 1. Schematic representation of the dissolution process for polymer molecules


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Thermodynamics background

The solubility of a given polymer in various solvents is largely

determined by its chemical structure. Polymers will dissolve insolvents whose solubility parameters are not too different from

their own. This principle has become known as like dissolves

like, and, as a general rule, structural similarity favors solubility.

Dissolution of an amorphous polymer in a solvent is governed by

the free energy of mixing

(1)

where Gm is the Gibbs free energy change on mixing, Hm is the

enthalpy change on mixing, T is the absolute temperature, and

Sm is the entropy change on mixing. A negative value of the free

energy change on mixing means that the mixing process will

occur spontaneously. Otherwise, two or more phases result from

the mixing process. Since the dissolution of a high molecular

weight polymer is always associated with a very small positive

entropy change, the enthalpy term is the crucial factor indetermining the sign of the Gibbs free energy change. Solubility

parameters were developed to describe the enthalpy of mixing.

Hildebrand pointed out that the order of solubility of a given

solute in a series of solvents is determined by the internal

pressures of the solvents.Later, Scatchard introduced the concept

of cohesive energy density into Hildebrands theories.

Hildebrand and Scott and Scatchard proposed that the enthalpy of

mixing is given by

(2)

where Vmix is the volume of the mixture, is the energy of

vaporization of species 1; V1 is the molar volume of species 1; and

1 is the volume fraction of 1 in the mixture. is the energy


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change upon isothermal vaporization of the saturated liquid to the

ideal gas state at infinite volume.

The cohesive energy, E; of a material is the increase in the

internal energy per mole of the material if all of the

intermolecular forces are eliminated. The cohesive energy density

(CED) Eq. (3), is the energy required to break all intermolecularphysical links in a unit volume of the material

(3)

where Hvap is the enthalpy of vaporization. The Hildebrand

solubility parameter is defined as the square root of the cohesive

energy density:

(4)

Eq. (2) can be rewritten to give the heat of mixing per unit volume

for a binary mixture:

(5)

The heat of mixing must be smaller than the entropic term in Eq.

(1) for polymersolvent miscibility (Gm 0): Therefore, the

difference in solubility parameters (1 - 2) must be small for

miscibility or dissolution over the entire volume fraction range.

However, these predictions with the Hildebrand solubility

parameters are made with the absence of any specific

interactions, especially hydrogen bonds. They also do not account

for the effects of morphology (crystallinity) and cross-linking. In

addition, there may be (non-ideal) changes with changes in

temperature and, in many cases, with changes in concentration.

One of the early schemes to overcome inconsistencies in the

Hildebrand solubility parameter introduced by hydrogen bonding

was proposed by Burrell, and is based on the assumption that

solubility is greatest between materials with similar polarities.

This method divided solvents into three categories depending on

the hydrogen bonding: poor, moderate, and strong hydrogen

bonding capabilities. The system of Burrell is summarized as


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follows: weak hydrogen bonding liquids are hydrocarbons,

chlorinated hydrocarbons and nitrohydrocarbons; moderate

hydrogen bonding liquids are ketones, esters, ethers, and glycol

monoethers; and strong hydrogen bonding liquids are alcohols,

amines, acids, amides, and aldehydes.

Hansen also accounted for molecular interactions and developedsolubility parameters based on three specific interactions.

The first and most general type of interaction is the non-polar,

also termed dispersive interactions, or forces. These forces arise

because each atom consists of negatively charged electrons

orbiting around a central positively charged nucleus. The moving

negative charges create an electromagnetic field, which attracts

all atoms to one another regardless of direction. All molecules

have this type of attractive force.

Polar cohesive forces, the second type of interaction, are

produced by permanent dipole dipole interactions. These polar

forces roughly correlate with the dipole moment of the

molecule.and the contribution to the dipole moment. They are

inherently molecular interactions and are found in most molecules

to one extent or another.

The third major interaction is hydrogen bonding. Hydrogenbonding is a molecular interaction and resembles the polar

interactions. These bonds are considerably weaker than covalent

bonds but are much stronger than ordinary dipoledipole

interactions.

Therefore, as Hansen proposed, the cohesive energy has three

components, corresponding to the three types of interactions:

(6)

Dividing the cohesive energy by the molar volume gives the

square of the Hildebrand solubility parameter as the sum of the

squares of the Hansen dispersion (D), polar (P), and hydrogen

bonding (H) components:


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Flory Huggins Theory

A thermodynamic theory of polymer solutions, first formulatedindependentlyby Flory and by Huggins, in which thethermodynamic quantitiesof the solution are derived from asimple concept of combinatorialentropy of mixing and a reduced

Gibbs energy parameter, the parameter.

Latice Model of a Polymer Solution

Boltzmann Equation


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According to the Boltzmann equation the number of ways thesolids and solvent molecules can be arranged is given by theequation.

Sm = K ln W (1)

where

Sm = entropy of mixing

K = Boltzmann constant

W = number of ways that one can accommodate the solventand solutemolecules arranged.

According to Flory Huggins theory, the entropy of mixing of

polymer solution is given by

Sm = -K [ Ns ln s +Np ln p] (2)

Where the subscript

S stands for the solvent

P stands for the polymer.

The volume fraction of s and p in the foregoing equations aregiven by.

s = (3)

p = (4)

Wheredenotes the number of solvent molecules

denotes the number of polymer molecules.

n is the number of polymer segments

The heat of mixing of polymer solution is given by


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Hm = K T NspWhere

- Chai is the Flory Huggins interacting parameter and whichis the measure of the solvent power.

The free energy of mixing of polymer solution in terms of isgiven by

Em = K T [ Ns p + Ns ln s + ln p] (5)

The Flory Huggins theory arrives at the equation for total freeenergy of dilution based on the following train of thought. Thereare two entropy factors which contribute to the free energy ofdilution: one arising from contact between the polymer segmentsand solvent molecules, and the other is called conformationalentropy. Thus the overall free energy of dilution will be the sum ofthe two terms as follows :

Fm = Fconform + Fcontact (6)

From thermodynamics

E conformation = R T [ln (1- p ) + (1-1/n) p] (7)

And E contact = R T [ p2 ]

(8)

So that Em = R T [ln (1- p ) + (1-1/n) p + p2 ] (9)

This equation is derived for a mono dispersed polymer. For apolydisperse systemthe n factor in the above equation has to be replaced with its

average value denoted as . In that case for a polydisperse

system equation (9) becomes.


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Em = R T [ ln (1- p )+ (1-1/ ) p + p2 ] (10)

in equation (10) the value of exceeds . some critical value thanthe system will tend to separate into 2 phases. in such casesvalue of n should be,

c = n + +

(11)

( p)c = 1+

(12)Here ( p)c represents the critical composition (i.e. critical volume

fraction of the polymer), corresponding to the critical value c

In spite of deriving Flory Huggins equation some short comingsare noticed.The most important correction has been done w.r.t the term

E contact = R T [ p2 ]

Above should be replaced by polynomial equation

(13)

Substitute equation (13) in equation (9) we get Flory Hugginsequation

Many parameters such as osmotic pressure, which could beexperimentally determined, may be arrived at from this theory.

The osmotic pressure is related to the free energy of dilution

through the following equation.

Where is the partial molar volume of the solvent.

On substituting for Fm from this equation in eqn (10)


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in dilute solutions, the value of p will be far less than unity andhence ln(1- p) can be written as follows

on appropriate substitution equation (19) will take the followingform [where in the osmotic pressure is related to molecular

weight ( ) and concentration (c) of the solution

Here partial specific volumeof the polymer.

equation 20 can be written as

or


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When the experimental osmotic pressure data is extrapolatde tozero concentration the higher power concentration terms can beneglected and the eqn gets reduced to

The term on the left hand side of eqn 24 indicates at 0

concentration or infinite dilution.

Limitations to the Flory-Huggins theoryThe Flory-Huggins theory is widely used still and has been

successful, largely, indescribing thermodynamics of polymer solutions. There are anumber of limitations tothe original formulation. The main onesare

Applicability only to solutions that are sufficientlyconcentrated that they have uniform segment density

There is no volume change of mixing (where as favorableinteractions between polymer and solvent molecules should

result in a negative volume change) There are no energetically preferred arrangements ofpolymer segments and solvent molecules in the solution

The interaction parameter, 12, is independent ofcomposition

Nature of Polymer Molecule in Solution

Due to their large number of carbon atoms bonded togetherforming a long chain, polymers can generally adopt a lot ofconformations. These conformations arise from the numerousinternal rotations that can occur through simple C-C bonds,originating a number of rotational isomers.

Nevertheless, although the rotation of each bond is able tooriginate different conformations, due to energy restrictions notall of them have the same probability of occurrence. In such a


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case, the most stable conformations predominate in solution, likeproteins and nucleic acids, that is in biopolymers mainly.

However, synthetic polymers particularly, can display a largenumber of possible conformations, and even though theseconformations have not the same energy, the differences aresmall enough so that the chains can change from oneconformation to another. This particularity gives a big flexibility tothe macromolecules, and due to this flexibility, the chains do notadopt a linear form in solution, but a very characteristicconformation, known as random coil.

Figure 4. The random coil model

Such a flexibility can be understood more clearly with the help ofmolecular models, as shown in Figure 5.

Figure 5. A C-C simple-bonded chain and its spacial

representation.

Let's assume C1, C2, and C3 are carbon atoms in the same plane.According to this, the atom C4 can occupy any place throughoutthe circle, which represents the base of a cone originated by therotation of the bond E3. The angles of such bonds are symbolizedby w, whereas the location of atom C4 is specified by the internalangle of rotation l.


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For a macromolecule in the solid state, the angle l has a fixedvalue due to the restrictions of the network packing. That is whythe possible rotational isomers do not occur. Nevertheless whenthis macromolecule is dissolved, the packing disappears and theangle l can vary widely, originating maximums and minimums ofenergy. Thus, the probability of reaching diverse stableconformations with each minimum of energy is high. On the otherhand, the variation of the internal angle of rotation is associatedto an energy change that, at minimums, is small. Hence, thechains can move freely to adopt such stable conformations. Thefact that the chains are changing from one conformation toanother is also favored, due to the low potential energy of thesystem. All these factors define, therefore, a flexiblemacromolecule and from these concepts, the typical random coilform arises.

You might ask if the "shape" or magnitude of the random coilwould remain the same once the polymer has been dissolved. Youwill find that the answer is absolutely negative and that thesituation will depend not only on the kind of solvent employed,but also on the temperature, and the molecular weight. Thepolymer-solvent interactions play an important role in this case,and its magnitude, from a thermodynamic point of view, will begiven by the solvent quality. Thus, in a "good" solvent, that is tosay that one whose solubility parameter is similar to that of the

polymer, the attraction forces between chain segments aresmaller than the polymer-solvent interactions; the random coiladopts then, an unfolded conformation. In a "poor" solvent, thepolymer-solvent interactions are not favored, and thereforeattraction forces between chains predominate, hence the randomcoil adopts a tight and contracted conformation.

In extremely "poor" solvents, polymer-solvent interactions areeliminated thoroughly, and the random coil remains so contractedthat eventually precipitates. We say in this case, that the

macromolecule is in the presence of a "non-solvent".

The particular behavior that a polymer displays in differentsolvents, allows the employ of a useful purification method,known as fractional precipitation. For a better understandingabout how this process takes place, lets imagine a polymerdissolved in a "good" solvent. If a non-solvent is added to thissolution, the attractive forces between polymer segments willbecome higher than the polymer-solvent interactions. At some


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point, before precipitation, an equilibrium will be reached, inwhich G = 0, and therefore H = TS, where S reaches itsminimum value. This point, where polymer-solvent and polymer-polymer interactions are of the same magnitude, is known as qstate and depends on: the temperature, the polymer-solventsystem (where H is mainly affected) and the molecular weight ofthe polymer (where S is mainly affected).

It may be inferred then, that lowering the temperature or thesolvent quality, the separation of the polymer in decreasingmolecular weight fractions is obtained. Any polymer can reach itsq state, either choosing the appropriate solvent (named qsolvent) at constant temperature or adjusting the temperature(named q temperature, or Flory temperature) in a given solvent.Table 2 compiles some values.

Polymer Solvent(s)q temperature

(oC)

Polyethylene n-Hexane 133

n-Hexanol / Xylene (70:30) 170

n-Octane 210

Polypropylene(atactic)

n-Butanol / Carbon Tetrachloride(33:67) 25

n-Butanol / n-Hexane (32:68) 25

Cyclohexanone 92

Polystyrene Benzene / n-Butanol (58:42) 35Cyclohexane 34-35

Cyclohexanol 79-87

Poly (vinyl acetate) Ethanol 19

Ethanol / Methanol (40:60) 36

Poly (vinyl alcohol) Ethanol / Water (41.5-58.5) 25

Water 97

Poly (vinyl chloride) Cyclohexanone 22

Dimethylformamide 36.5

Polyacrylamide Methanol / Water (2:3) 20

Polymethylmethacrylate Acetone -126

Cyclohexanol 77.6

Toluene -65

Dioxane / Water (85:15) 25

Table 2. q solvents for selected polymers.

The q temperature is a parameter arisen from Flory-Krigbaumtheory. It is used to calculate the free energy of mixing of a


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polymer solution in terms of the chemical potentials of thespecies. We will further study the q temperature relationship withother important parameters that characterize dissolved polymers.

So far we have analyzed the influence of the solvent and thetemperature in the dimensions of the random coil. However isequally important to know what happens to the viscosity of themacromolecular solution as the solvent becomes poorer.Considering the chain molecules as rigid spheres, when a changefrom a "good" solvent to a "poor" solvent takes place, the spheresbecome contracted. According to the Einstein equation, therelative viscosity hr is obtained from:

[4]

That is to say, dividing the viscosity of the solution (h) by theviscosity of the solvent (hs). From equation [4] it can be noticedthat hs is directly proportional to the volume fraction f that thesespheres occupy. Since, with the necessary considerations, thisreasoning can be transferred to macromolecules, which are notrigid spheres, it may be inferred that if the segments arecontracted in a "poor" solvent, the viscosity of the solution will besmaller. Therefore, viscosity can be adjusted according to thesolvent quality.

Temperature, however, will not affect the viscosity of a polymersolution in a relatively "poor" solvent. In this case, it should beconsidered that as the temperature increases, the viscosity of thesolvent (hs) decreases. However, on the other hand, when thetemperature is raised, a greater thermal energy will be granted tomolecules. Consequently, these molecules will tend to expandthemselves, increasing their volume fraction (f). Thus both effectsare compensated, and for this reason the change of viscosity due

to the increase of the temperature, is not significant.

The measurement of viscosity in dilute macromolecular solutionshas a fundamental importance not only in the determination ofmolecular weights, but also, as we will discuss later, in theevaluation of key parameters for the understanding of theconformational characteristics of polymer solutions.

Statistical Parameter


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According to what we have been studying so far, dissolvedpolymer molecules do not remain fully extended in a stationarystate; instead they adopt a typical random coil form in continuousmotion, changing readily from one conformation to another. Whenrotation around C-C simple bonds is hindered, the random coilconformation is reached only at high temperatures, due to thethermal energy conferred to the segments.

So you can ask the following question: it is possible to calculatethe size of a macromolecule in its typical random coil form, whosesegments are constantly changing from one conformation toanother? The answer is affirmative, only if such a size isexpressed in terms of statistical parameters, which represent anaverage of all the possible conformations. To that end, there aretwo very useful statistical parameters:

End-to end distanceRepresents the average distance between the first and the lastsegment of the macromolecule, and ranges between a maximumvalue and a minimum value. The maximum value appears whenchains are fully extended, in a planar, zigzag configuration knownas "all-trans", where the contour length can be calculated easily.The minimum value corresponds to the sum of Van der Waalsradii in each end.

Figure 6. Maximum value (left) and minimum value (right) for the

end-to-end distance r

The size of the macromolecule is given, in statistical terms, by the mean-squareend-to-end distance, (r)2. Other authors express the root mean-square end-to-end

distance, that is to say, ((r)2). The magnitude (r)2 is defined according to:

(5) Where W is a probability distributionfunction.


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The calculation of the mean-square end-to-end distance (r)2,varies according to the chain type, and the interactions that weretaken into account.

Let's consider the simplest model of a polymer chain, i.e. an idealpolymer, consisting of a series of N segments of length L. Let'sassume that the chain segments are bonded according to a linearsequence, without any restriction regarding bonding angles w andinternal angles of rotation l (Figure 5), so that the atoms areseparated each other at fixed distances but located in anydirection. Thus, the calculation of (r)2 can be made by means of aprocedure known as random flight. According to this procedureand following a mathematical reasoning equation [5] can be re-written as follows:

(6)

Where subscript f indicates that a random flight approximation isbeing considered, originating a model known as freely jointedchain.

Nevertheless, the situation is somewhat more complicated whentrying to transfer the calculation of (r)2 to real macromolecules,where restrictions indeed exist, and therefore chain atoms cannot

be located in any direction. Factors like solvent type, chain type,and the groups attached to the polymer backbone, do causeinteractions, generating deviations from the freely jointed chainmodel. For this reason (r)2 is higher than that obtained bycalculation of the random flight. These interactions can be dividedin: short range interactions and long range interactions.

Short range interactions are related to the structuralcharacteristics of the macromolecule, considering bond types andthe interactions between segments or neighboring atoms. These

factors originate steric repulsions, which limit the values of theinternal angles of rotation (figure 5) since in such a case, they arenot all equally probable. The magnitude of this effect is related tothe size of the substituent groups. Hence the random coil willexpand itself, in order to avoid such repulsions.

This model is known as unperturbed dimension, since neitherinteractions between non-neighboring chain segments nor solventinteractions (long range interactions, to be discussed later) are


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being considered. To this end, the mean-square end-to-enddistance of the unperturbed dimension, (r)2o, is expressed asfollows:

(7)

Where (r)2fr represents the mean-square end-to-end distance ofthe free rotation chain, that is to say, under the condition that thebonding angles w (figure 5) remain fixed, independently of thepresence of substituent groups. The s factor, referred to asconformation factor, is a parameter related to the impediments torotation that real chains show, compared to that with a free chainrotation. The s factor depends on temperature and sometimes onthe solvent, and offers interesting information about theconformation of a certain macromolecule and usually, it is

increased in the presence of bulky groups.

Although the corrections introduced by the short rangeinteractions offer a more approximate description of realmacromolecules in dilute solutions, such interactions do notcontemplate the behavior of non-neighboring chain segments,each one occupying a certain volume from which all the othersegments are excluded. Such effect, known as excluded volumeand its influence on the dimensions of the macromolecular chains,

has been the subject of numerous studies for a long time. Itstheoretical calculation has been carried out by means of statisticand the aid of computer simulation.

To this end, in order to make possible the calculation of the end-to-end distance considering the excluded volume effects, the longrange interactions have been introduced, which consider bothsolvent interactions and interactions between atoms or non-neighboring segments. One may speculate that the long rangeinteractions can produce a bigger chain expansion over its

unperturbed dimensions, since now, due to the excluded volumeeffect, such conformations where two remote segments canoccupy the same space at the same time should be eliminated.Hence, the end-to end distance is given by:

(8)


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Where a is the linear expansion factor, and (r)2o the end-to-enddistance of the unperturbed dimension.

The value of a depends on the number of segments, thetemperature, and the solvent nature. When a values are high, it issaid that the solvent is thermodynamically "good". Therefore,remembering what we mentioned in previous pages, polymer-solvent interactions will be favored, making the random coil to beexpanded and allowing the excluded volume effect to occur. Onthe contrary, when a values are low, the solvent is consideredthermodynamically "poor" and the random coil will be contracted,in order to minimize the contact with its segments.

Analyzing equation [8], it can be noticed that when a = 1, asituation that can be reached with a solvent sufficiently "poor" orat low temperatures, the dimensions of the macromolecule,

affected by long range interactions, matches those of theunperturbed dimension. Under such conditions, the polymer-solvent and polymer-polymer interactions are of the samemagnitude, and that particular "q state", which we mentionedearlier, is reached. If such a condition was reached at a giventemperature, this temperature will be considered as "qtemperature" or "Flory temperature"; if it was reached in a givensolvent at T = a, this solvent will be considered as a "a solvent".

By means of the preceding discussion, the relevance in theevaluation of the conformation factor s, and the linear expansionfactor a, while studying the conformational properties of polymersin dilute solutions, is clearly demonstrated. While s is related tothe geometry of the chains and its structural characteristics(considered by the short range interactions) a measures themagnitude of the excluded volume effect as well as polymer-solvent interactions, described through the long rangeinteractions. The experimental determination of factors s and a,will be discussed briefly in the next section.

Radius of GyrationIn the case of branched chains, with a large number of ends, itturns out more appropriate to talk about the square-mean radiusof gyration instead of the end-to-end distance, which is moreapplicable to linear polymers. Represented as (s)2, the radius ofgyration is the square mean radius of each one of the elements ofthe chain measured from its center of gravity. Although (s)2 isdefined according to:


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(9)

Being N the number of segments, and (h i)2 the square mean

distance of i-th segment from the center of gravity, its value is

often related to the square mean distance (r)2 through a constant,in the generic form:

(10)

Thus, it can be noticed that the dimensions of a branched chainare smaller than those of a linear chain with the same molecular

weight, or the same number of segments. This simple relation canbe applied to the three situations considered in a), and if weexpress (s)2 in terms of equations [6[, [7], and [8] we will obtain,respectively, the value of the square-mean radius of gyration for a"freely joined chain" (equation [11]), an "unperturbed chain"(equation [12]) and a chain being affected by long rangeinteractions (equation [13]):

(11)

(12)

(13)

The following schematic representation of a random coil, showsthe differences between the end-to-end distance (r) and radius ofgyration (s).


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Figure 7.

Theta Solvent

Definition: A solvent which performs in an ideal manner (activity

coefficient = 1) in dilute solution measurements of molecular

weight.

In a polymersolution, a theta solvent (or solvent) is a solvent

in which polymer coils act like ideal chains, assuming exactly their

random walk coil dimensions. Thermodynamically, the excess

chemical potential of mixing between a polymer and a thetasolvent is zero

Physical Interpretation

The conformation assumed by a polymer chain in dilute solutioncan be modeled as a random walk ofmonomer subunits using afreely jointed chain model. However, this model does not accountfor steric effects. Real polymer coils are more closely represented
http://en.wikipedia.org/wiki/Polymerhttp://en.wikipedia.org/wiki/Solutionhttp://en.wikipedia.org/wiki/Solventhttp://en.wikipedia.org/wiki/Ideal_chainhttp://en.wikipedia.org/wiki/Random_walkhttp://en.wikipedia.org/wiki/Chemical_potentialhttp://en.wikipedia.org/wiki/Chemical_structurehttp://en.wikipedia.org/wiki/Monomerhttp://en.wikipedia.org/w/index.php?title=Freely_jointed_chain_model&action=edit&redlink=1http://en.wikipedia.org/wiki/Steric_effectshttp://en.wikipedia.org/wiki/Polymerhttp://en.wikipedia.org/wiki/Solutionhttp://en.wikipedia.org/wiki/Solventhttp://en.wikipedia.org/wiki/Ideal_chainhttp://en.wikipedia.org/wiki/Random_walkhttp://en.wikipedia.org/wiki/Chemical_potentialhttp://en.wikipedia.org/wiki/Chemical_structurehttp://en.wikipedia.org/wiki/Monomerhttp://en.wikipedia.org/w/index.php?title=Freely_jointed_chain_model&action=edit&redlink=1http://en.wikipedia.org/wiki/Steric_effects


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by a self-avoiding walk because conformations in which differentchain segments occupy the same space are not physicallypossible. This excluded volume effect causes the polymer toexpand.

Chain conformation is also affected by solvent quality. Theintermolecular interactions between polymer chain segments and

coordinated solvent molecules have an associated energy ofinteraction which can be positive or negative. For a goodsolvent, interactions between polymer segments and solventmolecules are energetically favorable, and will cause polymercoils to expand. For a poor solvent, polymer-polymer self-interactions are preferred, and the polymer coils will contract. Thequality of the solvent depends on both the chemical compositionsof the polymer and solvent molecules and the solutiontemperature.

If a solvent is precisely poor enough to cancel the effects ofexcluded volume expansion, the theta () condition is satisfied.For a given polymer-solvent pair, the theta condition is satisfiedat a certain temperature, called the theta () temperature ortheta point. A solvent at this temperature is called a thetasolvent.

In general, measurements of the properties of polymer solutionsdepend on the solvent. However, when a theta solvent is used,

the measured characteristics are independent of the solvent.They depend only on short-range properties of the polymer suchas the bond length, bond angles, and sterically favorablerotations. The polymer chain will behave exactly as predicted bythe random walk or ideal chain model. This makes experimentaldetermination of important quantities such as the root meansquare end-to-end distance or the radius of gyration muchsimpler.

Additionally, the theta condition is also satisfied in the bulkamorphous polymer phase. Thus, the conformations adopted bypolymers dissolved in theta solvents are identical to thoseadopted in bulk polymer.

Theta Temperature

Definition: With respect to molecular interactions in dilute

polymer solutions, theta temperature is the temperature at which
http://en.wikipedia.org/wiki/Self-avoiding_walkhttp://en.wikipedia.org/wiki/Excluded_volumehttp://en.wikipedia.org/wiki/Intermolecular_interactionshttp://en.wikipedia.org/wiki/Ideal_chainhttp://en.wikipedia.org/wiki/Root_mean_squarehttp://en.wikipedia.org/wiki/Root_mean_squarehttp://en.wikipedia.org/wiki/Radius_of_gyrationhttp://en.wikipedia.org/wiki/Amorphoushttp://en.wikipedia.org/wiki/Phase_(matter)http://en.wikipedia.org/wiki/Self-avoiding_walkhttp://en.wikipedia.org/wiki/Excluded_volumehttp://en.wikipedia.org/wiki/Intermolecular_interactionshttp://en.wikipedia.org/wiki/Ideal_chainhttp://en.wikipedia.org/wiki/Root_mean_squarehttp://en.wikipedia.org/wiki/Root_mean_squarehttp://en.wikipedia.org/wiki/Radius_of_gyrationhttp://en.wikipedia.org/wiki/Amorphoushttp://en.wikipedia.org/wiki/Phase_(matter)


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the second virial coefficient disappears. That is, the temperature

at which the coiled polymer molecules expand to their full contour

lengths and become rod-shaped. Also known as FLORY

TEMPERATURE.

VISCOSITY OF DILUTE POLYMER SOLUTIONS

The viscosity of a dilute polymer solution depends on severalfactors namely, the nature of the polymer and that of the solventtheir concentrations, the polymer molecular weight thetemperature and the shear rate.

At the molecular level, the viscosity of a polymer solution isdirect measure of the hydrodynamic volume of the polymer

molecules.Hydrodynamic volume is the apparent volumeaccupied by the expanded or swollen molecular coil along withthe imbibed solvent and can be defined in terms of the expansionfactor and unperturbed end-to end distance ((ro)2).Polymer willexhibit a higher viscosity in a good solvent than in a poor solventand that , in the same solvent, the viscosity eill be directlyproportional to the molecular weight.

Reference

1. POLYMER SCIENCE


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By V.R.Gowariker, N.V.Viswanathan, Jayadev Sreedhar

2. POLYMER SOLUTIONS

by IWAO TERAOKA

3. ESSENTIALS OF POLYMER SCIENCE AND ENGINEERING

By Paul C. Painter, Michael M. Coleman

4. TEXT BOOK OF POLYMER SCIENCE

BY Fred W. Billmeyer

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Polymer Solution