ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS phần 4 pptx

Variation of [η] against molar mass in the case of polystyrene solutions in a good solvent and under θ conditions.. molar mass, which was subsequently proved true for rigid polymers Figu

MACROMOLECULAR MASSES AND SIZES BY SCATTERING TECHNIQUES 181

to be scattered in a solid angle  per unit volume of the sample The total scattering

section can be split into two contributions, one coherent and the other noncoherent:

b takes positive values for isotopes scattering out of phase and negative values for

in-phase scattering centers Thus, for hydrogen we have

b H = −0.374 × 10−12 cmand, for deuterium,

Recalling that in light scattering, the “contrast factor” K is written as

where r e is the radius of an electron (2.81× 10−15m) and Z is the difference

between the number of electrons of constitutive polymer atoms and those of solvent

Z can be deduced from the electron density of the various components, ρe,u

(repetitive unit) and ρe,s (solvent):

that can be used to determine M w , A2, s2 Due to their very short wavelength,

neutrons and X-rays serve to measure small radii of gyration

In order to get the same effect, the quantity corresponding to P (θ) must be the

same for light scattering (λ = 450 nm) or X-ray (λ = 0.1 nm) scattering to avoid

the effects due to interferences because they are detectable for sizes smaller thanthose measured in light scattering, that is, when s2 1/2 λ0/20 This imposesthat experiments be carried out at very small angles (θ 2◦); thus in the twolatter cases the detector has to be placed sufficiently away ( 1 m) from the sam-

ple so that there is enough separation between the incident and the scatteredbeams

On the other hand, these scattering techniques give information at a scale muchsmaller than that resulting from light scattering by photons; thus one can measureradii of gyration s2 1/2 of [1− 92(<s2)/3] much smaller macromolecules

At these small angles, P (θ) can be approximated by the Guinier function

[1− q2 (s2 /3)], which allows the determination of the radius of gyration of

macromolecules independently of their shape; the higher limit for measurements

by SAXS, SANS, and light scattering is 5 nm, 20 nm, and 200 nm, respectively.Synchrotron sources of radiation generate X-rays, whose wavelengths vary from

0.06 to 0.3 nm With laboratory equipment, Kα radiation of Cu is used with awavelength of 0.154 nm

Neutrons are produced by nuclear reactors and have to be slowed down in order

to obtain “cold” neutrons of about 1 nm wavelength

MASS SPECTROMETRY APPLIED TO POLYMERS 183 6.3 MASS SPECTROMETRY APPLIED TO POLYMERS

Mass spectrometry is useful for the determination of molar masses of simplemolecules It is based on the vaporization and the ionization of the entities to

be studied Sent into an electric (or magnetic) field, each of these charged species

undergoes a deflection proportional to its m/z ratio, where m represents the mass

of the particle and z represents the number of charges carried.

Because of their thermal instability and their lack of volatility, polymers cannot

be characterized by this technique

The use of “MALDI” (Matrix-Assisted Laser Desorption Ionization) equipped with a “time of flight” (TOF) spectrometer offers an elegant and effective method for

the precise and absolute determination of polymer molar masses Initially designedfor the characterization of biomolecules, MALDI-TOF spectrometry is now exten-sively applied to synthetic polymers; in this technique the polymer to be studied isdispersed in an organic matrix that is volatilized under the effect of a laser radiationwhose wavelength is in the absorption range of the matrix Owing to the specificmolecular interactions between polymer and matrix— which plays an important role

in the chain desorption—each polymer requires a suitable matrix For example,1,8,9-trihydroxyanthracene is well suited to poly(methyl methacrylate)

In order to avoid interchain entanglements, it is essential experimentally to keepthe polymer concentration in the matrix very low Upon volatilization of the matrix,the polymer species are desorbed and ionized (by a mechanism still ill-known),which makes them sensitive to the accelerating effect of an electric field Forpolymer analysis, the positive mode —the one which causes the formation of apositive charge on the chain by interaction with a cationic species (proton or metalcation)—is generally used

In order to obtain simple and easy to analyze spectrograms, chains should bemonocationized The separation of polymers (pol− H+ or pol− Met+) according

to their molar mass is obtained through the time of flight necessary for each ofthe species to travel from the target to the detector Although molar masses up

to 1.5× 106 g·mol−1 could be measured, most of the published studies pertain to

polymers of molar mass ranging between 1000 and 2× 104g·mol−1; this technique

is an extremely sensitive one since quantities of matter in the range of femtomolescan be detected

Problems encountered in MALDI-TOF mass spectrometry are due to:

• The low resolution beyond molar masses of about 2.5× 104g·mol−1

• The difficulty to “cationize” certain nonpolar polymers

• The difficulty to desorb macromolecules of high molar masses

Remark The upper limit of molar masses for well-resolved signals depends

on the molar mass of the monomer unit Beyond that limit, an envelope ofthe constituting signals is observed

The example chosen to illustrate this technique of characterization is a sample

of a macrocyclic polystyrene cyclized through an acetal function (M n= 6900 and

M w /M n= 1.5) and represented here:

-O

O

O O

~~PS~~

(n−5)

In the MALDI-TOF spectrogram shown in Figure 6.12, the exact degrees of

polymerization (n) of the chains present in the samples can be seen without

ambi-guity through their mass and the various molecular groups carried by the chainsidentified; each signal corresponds to chains whose molar mass is given by therelation

7031.8

7447.7 7655.3 7863.2 8072.0 8279.8 8487.0 8694.7 6095.5

Figure 6.12 MALDI-TOF mass spectrogram of a macrocyclic polystyrene whose formula is

given above.

VISCOSITY OF DILUTE SOLUTIONS— MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 185 6.4 VISCOSITY OF DILUTE SOLUTIONS — MEASUREMENT

OF MOLAR MASSES BY VISCOMETRY

Molar masses and molecular dimensions of polymers are accessible not onlythrough scattering techniques but also from viscosity measurements Indeed, theresponse of macromolecules to the application of hydrodynamic forces can giveinformation about their volumes and their dimensions and thus indirectly abouttheir molar masses By definition, the viscosity of a liquid is proportional to theproduct of the flow time of a characteristic volume times its density:

The simplest case corresponds to a flow behavior described by the Newton law,

in which the viscosity (η) is the ratio of the shear stress (σ) to the shear rate (˙γ)

and more specifically the slope of the straight line drawn from the variation of theshear stress versus the shear rate (η = σ/˙γ) Liquids exhibiting such behavior are

called Newtonian, and their viscosity is independent of the shear rate

Remark. Non-Newtonian liquids do not have this linear variation of σ

versus ˙γ, in the entire range of shear rates

Except for very high molar mass samples, viscosities of dilute polymer solutionsare Newtonian, and the relations between the polymer molar mass and its viscosityare established in this context

More than the proportionality constant, which relates the viscosity to the flowtime, experimenters are interested in comparing the viscosity of a polymer solutionwith that of pure solvent (η1) This has given rise to the notion of relative viscosity(ηr= η/η1), specific viscosity [ηsp= (η − η1)/η1 = ηγ−1], or reduced viscosity

(ηred= ηsp/c2).

The specific viscosity is an indicator of the increase in viscosity due to theaddition of a polymer, whereas the reduced viscosity characterizes the propensity

of a given polymer to increase the relative viscosity and is also called intrinsic

viscosity [η] in the limit of infinite dilutions:

Thus, intrinsic viscosity has the dimension of a specific volume

6.4.1 Variation of Viscosity with Concentration

The viscosity of a dispersion of sufficiently diluted rigid particles —so that theireffect is thus simply additive —can be described by the Einstein viscosity relation:

η = η1[1+ B12+ B22· · ·] (6.43)

Figure 6.13 Example of extrapolation to zero concentration of the variation of η sp/c2 and ln( ηr /c2 ) against concentration.

with B1= 2.5 when particles are nonsolvated rigid spheres and B2= 14.1; in this

equation, B1 and B2 can take different values, depending on the shape and the size

of the particles under consideration

In the previous expression the volume fraction of the particles 2 = V2/V1 can

be replaced by their mass concentration c2, after observing that V2 is equal to N2 VH(i.e., the product of the number of particles times their hydrodynamic volume) and

that the concentration (c2) and the number (N2) of these particles in the volume (V ) are related by

where [η] is equal to 5/2(VHNa/M) and kH, which is called the Huggins coefficient,

is equivalent to 4B2/25 kHcan be obtained by plotting the linear variation ofηsp/c2

against c2 kHtakes values close to 1/3 when the polymer is in a good solvent and

can grow up to 0.5–1 if a bad solvent is used kHis thus a criterion of the quality

of solvent

6.4.2 Relation Between Viscosity and Molar Mass of a Polymer

Long ago, Staudinger had the intuition that the molar mass of a polymer and itsviscosity must be related and thus postulated that [η] must be proportional to the

VISCOSITY OF DILUTE SOLUTIONS— MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 187

0.74

0.50 [h] (ml/g)

Figure 6.14 Variation of [η] against molar mass in the case of polystyrene solutions in a good solvent and under θ conditions.

molar mass, which was subsequently proved true for rigid polymers (Figure 6.14).Actually, molar mass and viscosity could be related in an empirical manner throughthe Mark–Houwink–Sakurada (M –H–S) equation:

where K and α are constants varying with the polymer, solvent, and temperature

under consideration The value taken by the exponent α gives information about

the conformation of the polymer in a given solvent and even its shape

Thusα is equal to 0 for spheres, 0.5 for statistical coils in a nonperturbed state

(θ conditions), ∼0.8 for chains in solution in a good solvent, and 2 for rigid rods

For polymers with a wormlike shape, the coefficientα is intermediate between that

of perturbed chains and rods —that is, between 0.8 and 2 Due to the non-Gaussiancharacter of small chains —which are not statistical coils —the “M –H–S” equationapplies strictly only to chains whose molar masses are higher than 2× 104g·mol−1.

The constants K andα for small chains in a given solvent are therefore different

from those determined for Gaussian chains with the same repetitive units

To know the type of average molar mass accessible by viscosity measurements,the following reasoning can be used; becauseηspis equal to [η]c2 within the limit

of low concentrations,ηspcan also be written as

The viscosity of a sample is thus the mass average of viscosities of the collection

of chains present in the medium Average molar masses obtained from viscosity

experiments are called viscosity average molar masses.

6.4.3.1 Case of Rigid Spheres As previously shown, the intrinsic viscosity

corresponds to a specific volume —that is, to the hydrodynamic volume of 1 g

of the polymer analyzed within the limit of infinite dilutions Insofar as relationscan be established between the hydrodynamic volume of a polymer of a givenconformation and its molar mass, use can be made of viscosity measurements

to determine the molar mass of a sample If the polymer analyzed is of sphericalshape, its hydrodynamic volume corresponds to the volume of an equivalent sphere

where R e and R i are the outer and inner radii of a partially hollow sphere; in our

case related to solid spheres, R i = 0 and R e ≡ Rsph; the conversion factor Q (Rsph

= Qsph,s) is thus equal to (5/3) 1/2 and then one obtains:

withφsph,s equal to 13.57× 1024mol−1

Knowing that the relation between the molar mass of a spherical object and itsradius of gyration is written as

s = (3/5)(4πρ)N 1/3

it is easily shown that the intrinsic viscosity of a sphere is independent of its molarmass and depends only on its density

6.4.3.2 Case of Statistical Coils The same reasoning—that is, identifying

a polymer (here a statistical coil) to an equivalent sphere of radius R — can be

VISCOSITY OF DILUTE SOLUTIONS— MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 189

the radius of gyration of such equivalent sphere and thus to calculate Q , which is

supposed to depend on the distribution of segments in the statistical coil

Free Draining Model In this model, known as the “Rouse model,” the polymer

is represented as beads interconnected by massless springs free of hydrodynamicinteractions The solvent passes freely through the statistical coil and exerts fric-tional forces on the segments, or the beads: each center or friction point movesindependently as if the other points do not exist The viscosity (η) is thus the prod-

uct of the frictional coefficientξ of a segment and a global factor F, which takes

into account not only the friction undergone by each of the X segments but also

the conformation effects:

F = (ρN a / 6)( s2 0/M)X

In dilute solutions, the density (ρ ≡ mcoil/Vcoil) can be considered as identical to

the concentration: C2 = m2/V , where m2is the mass of polymer and V its volume.

Viscosity can be written as

η = η1( η/η1) ∼= η1(η − η1)/η1= η1ηspand

With s2 0/M being constant, it is included in Kη

Thus, the free draining model predicts that the exponent α of molar mass is

equal to 1 for chains without excluded volume as in the Staudinger empiricalformula Except for a few cases, this model is not very realistic because it neglectshydrodynamic interactions between elements of chains

Unperturbed Statistical Coils In the Kirkewood and Riseman model, the

poly-mer is represented as a collection of beads interconnected by bonds of length L

and interacting with each other This method involves the calculation of the bations due to the interactions between repeating units and to the long-range onesinduced by the chains Using the Oseen formula, these authors obtained

Through the function Q · f (Q), this model describes both the case of the free flow

and that of impermeable chains Depending on the degree of friction between the

solvent and the chain, Q · f (Q) varies from values close to 0, in the case of free

draining chains, to 1.26 for impermeable chains In the latter case, φ is equal to

4.22× 1024mol−1 Because the exponentα is not supposed to vary with the molar

mass in this approach, it is simply assumed that the degree of permeability of

the chains is related to their molar mass The product ( s2 0/M )3/2 then varies

with M

Statistical Coils in Perturbed Mode This model is also called the Flory– Fox

model In the previous model, the effect of excluded volume was not taken intoconsideration According to the Flory and Fox analysis, the latter model appliesonly to the case of unperturbed chains underθ conditions; the case of chains in a

good solvent requires a separate treatment The Flory– Fox model is based on theassumption that long-range interactions and the perturbations that they cause donot modify the flow of a solution:

also independent of the molar mass of the sample Since Qθ= R H,θ/s0= 0.87 for

such nonperturbed chains (in the case of spheres, Qsph= (5/3)1/2 as it was shownpreviously),φθis equal to 4.22× 1024mol−1, which corresponds to the value deter-mined by Kirkewood and Riseman for impermeable chains Figure 6.15 shows how

φ varies in reality

To take into consideration the continuous increase of the exponentα (in [η] =

KMα) with the molar mass in a good solvent, Flory and Fox conditioned the degree

of permeability of the chains to their mass, and they did it through the excluded

volume effect Considering that for perturbed chains s varies with M2 as

s2 1/2 = KM 3/5

(6.53)

VISCOSITY OF DILUTE SOLUTIONS— MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 191

10

Φ 10−24 (mol−1)

4,22

10 3 10 4 10 5 10 6 10 7 Mw ( g /mol)

Figure 6.15 Variation of , considered as an ‘‘universal’’ constant or Flory constant, versus

the molar mass: case of polystyrene in good solvent ( ) and under θ conditions ( ).

which gives the following expression for the intrinsic viscosity,

[η] = φK3M 9/5 M−1= Kη M 4/5 (6.54)

In the latter model α can take values close to 0.8 for flexible chains in a good

solvent According to the de Gennes theory, the exponent in the relation betweenthe molar mass and the radius of gyration is equal to 0.588 for chains with excludedvolume and thusα is equal to 0.764 Because the phenomenon of excluded volume

becomes more significant at higher molar masses, there is no universal value ofφ

for perturbed chains; thusφ differs from one system to another and decreases as the

molar mass of the chains grows The radius of gyration increases by a factorα in

a good solvent according to the relationαs= [ s2 / s2 0]1/2; the hydrodynamicradius undergoes the same phenomenon and one obtains αH= [VH/VH,θ1/3 Thepreceding equation can thus be written as

6.4.3.3 Cases of Rods Examples of rigid rods among synthetic polymers are

those which adopt a helical conformation either due to the size of their substituents

[poly(triphenyl methacrylate), etc.] or to internal attraction forces [poly(L-γ-benzyl

glutamate), etc.] The volume of such cylindrical rods is the product of their length

L times the square of their radius R times π, that is,

V = πR2L

For such shapes, the radius of gyration is defined as

s2=

 L/2 0

6.4.3.4 Case of Branched Polymers Due to their branching points, branched

polymers are characterized by hydrodynamic dimensions smaller than those of theirlinear counterparts Their compactness can be assessed through the comparison oftheir radius of gyration with that of linear equivalents of the same molar massunderθ conditions,

g=s2 0, branched

s2 0, linear

(6.58)

The g parameter is thus always lower than 1 Because branched polymers are

characterized by θ temperatures lower than those of their linear counterparts, the

determination of s2 0 by scattering techniques underθ conditions is not easy

Comparing the radii of gyration determined in a good solvent is not ily the appropriate solution because branched and linear polymers swell differ-ently Indeed, the expansion coefficient (α) varies with the type of structure and

necessar-αbranched<αlinear In the case of star polymers, the closer to the core, the morestretched are the branch segments and at the same time a star polymer as a wholecannot swell as much as its linear equivalent

Actually, viscosity measurements provide an easy means to compare intrinsicviscosities of polymers and to determine the compactness of branched ones Under

of two branches to a six-arm star on its hydrodynamic volume and hence on its

VISCOSITY OF DILUTE SOLUTIONS— MEASUREMENT OF MOLAR MASSES BY VISCOMETRY 193

[η]star; such a 33% increase (equivalent to two more branches) in the molar mass

in a linear polymer would have caused a subsequent increase in [η]lin.

Two additional branches

Various expressions predicting the variation of gwith the number of branches (f )

have been proposed The Fixman–Stockmayer model gives

where g can be one of the functions described by the two previous models sassa proposed a slightly modified version of the same approach that provides the

Cas-best agreement with experiments Gnow becomes

G= [Y ... prior establishment of a calibra-tion curve from elution volumes of standard samples as a function of their molarmass Standard samples of polystyrene for polymers soluble in organic solvents

or...

a solvent or of an inorganic support (silica) filling a column

Straightforward and easy to handle, SEC provides first-hand information aboutthe distribution of molar masses of the sample... equilibrium occurs in spite of the flow of the mobile phasebecause of the fast diffusion of the solute molecules in and out of the stationary

phase Under standard conditions, the equilibrium