Single chain diffusion process and proton magnetic relaxation in polymer melts

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Single chain diffusion process and proton magnetic relaxation in polymer melts

J.P. Cohen-Addad

To cite this version:

J.P. Cohen-Addad. Single chain diffusion process and proton magnetic relaxation in polymer melts.

Journal de Physique, 1982, 43 (10), pp.1509-1528. �10.1051/jphys:0198200430100150900�. �jpa-

00209533�

Single ^{chain diffusion process}

and proton magnetic relaxation in polymer ^melts

J. P. Cohen-Addad

Laboratoire de Spectrométrie Physique (*), Université Scientifique ^et Médicale de Grenoble,

B.P. 53X, 38041 Grenoble Cedex, ^France

(Reçu ^{le 17} février 1982, révisé le 25 mai, accepté ^{le 28} juin 1982)

Résumé.

Le but du présent ^{travail est} de relier les propriétés de cohérence quantique ^{des moments} magnétiques

de protons attachés à des chaînes polymères fortement enchevêtrées aux propriétés dynamiques de basses fré- quences des liquides polymères. ^{Les résultats} théoriques ^et expérimentaux démontrant la présence d’une interaction tensorielle de spins, résiduelle, induite soit par l’effet de contrainte topologique ^{sur les} chaînes, soit par des ponts covalents sont commentés à partir d’un modèle de macromolécule linéaire à enchaînement libre ; chaque ^liaison

est supposée porter ^une paire ^de protons. ^Le concept de sous-chaîne est utilisé. On admet que chaque sous-chaîne,

de longueur ^{de contour} Le, possède ^{un vecteur} bout-à-bout temporaire re, dû ^aux contraintes topologiques ^exer-

cées sur la chaîne; ^{à ce vecteur} bout-à-bout est associé un ordre orientationnel moyen ^et temporaire ^{des unités} monomères; ^{celui-ci est} observé par la RMN. Le signal d’induction libre G(03B4t) de l’aimantation est relié de façon générale ^aux propriétés ^des polymères par les relations : i) ^{03B4 ~} 0394G l/Le, pour les chaînes fortement enchevêtrées ;

l’étant la longueur statistique ^{de Kuhn et} 0394G le ^{second moment} de la raie de résonance observée dans l’état vitreux.

L’ordre de grandeur ^{de 03B4-1 est en} bon accord avec les temps de croissance observés sur les chaînes réelles;

ii) 03B4g ^~ q2/3 0394G/Le4/3, pour les gels polymères; q ^{est le} rapport ^de gonflement ^du gel; ^la dépendance Le-4/3 ^de 03B4g ^en fonction de la longueur de la chaîne élémentaire est associée à l’absence de fluctuations de grande amplitude

des points ^de liaison covalente. Une expression générale ^de G(03B4g ^t), incluant le rapport d’élongation 03BB ^du gel, ^est

aussi présentée.

Les effets du processus de désenchevêtrement rapide ^{sur le} signal d’induction libre de l’aimantation nucléaire sont décrits en considérant que le couplage ^de spin résiduel, 03B4, associé à l’ordre orientationnel moyen, temporaire, ^des

unités monomères est détruit selon un processus de rétrécissement par le mouvement. On suppose que le ^mouve- ment d’une chaîne polymère peut être décrit selon un modèle de Rouse, ^bien qu’il ^{soit connu} que certaines pro-

priétés viscoélastiques ^sont décrites de manière plus ^{exacte à} partir du modèle de ^« reptation »; cependant ^le

modèle de Rouse est bien approprié à l’illustration des propriétés ^{RMN : il} montre comment 03B4-1 pourrait ^être

utilisé comme une référence de temps ^assez longue pour explorer ^les temps de renouvellement de configurations ^de

chaînes. Une comparaison qualitative ^avec les résultats obtenus à partir du modèle de désengagement ^{de chaîne} proposé par De Gennes ^est aussi donnée dans le présent ^travail.

Abstract.

The present ^{work aims at} connecting properties ^{of the} quantum coherence of proton magnetic ^mo-

ments linked to strongly entangled polymer ^{chains to low} frequency dynamical properties ^of polymeric liquids.

Theoretical and experimental ^results giving evidence for the presence of ^a residual tensorial spin coupling ^induced by chain confinement or crosslinks are commented by using ^a freely jointed chain model where every link is sup-

posed ^to carry ^a proton pair. ^The submolecule concept ^is applied. ^{Due to chain} confinement, every submolecule of

contour length Le, ^is supposed ^{to have a} temporary end-to-end vector re, inducing ^a temporary average orienta- tional order of monomeric units, observed from NMR. The free induction decay G(03B4t) ^{of the} magnetization ^is given

a general connection to polymer properties through the relations : i) ^{03B4 ~} 0394G l/Le, ^for strongly entangled chains;

with l, ^{the Kuhn} step length and 0394G the second moment of the resonance line observed in the glassy ^{state. The order}

of magnitude ^{of 03B4-1 is in} good agreement ^with decay ^times actually ^{observed on real} chains; ii) 03B4g ~ q2/3 0394G/Le4/3,

for polymeric gels; ^with q, the swelling ratio of the gel; ^the Le-4/3 dependence upon elementary ^chain length ^is

shown to be related to the absence of any large amplitude fluctuations of crosslink points. ^A general expression ^of G(03B4g t) including ^the stretching ratio, 03BB, ^{of the} gel ^{is also} given.

Effects of a fast disentanglement process ^on the free induction decay ^{of the} magnetization ^{are described} by ^consi- dering that the residual spin coupling, 03B4, associated with the temporary average orientational order of monomer

units is destroyed according ^{to a motional} narrowing process. The chain motion is supposed ^{to occur} according

Classification

Physics ^Abstracts

61.40K - 76.60E

(*) ^{Associe au C.N.R.S.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198200430100150900

to a Rouse model. This ^was chosen for the sake of simplicity, although ^some viscoelastic properties ^{are known to}

be described in a righter way from the so-called « reptation » ^{model; however,} the Rouse model is well appropriate

to the illustration ofNMR properties : it shows how 03B4-1 might serve as a long reference time scale for investigations

of renewal times of chain configurations. ^A qualitative comparison with results obtained from the model of chain

disengagement process proposed by ^De Gennes, ^is given.

1. Introduction.

Fluctuation properties ^{of disor-}

dered systems ^like polymeric liquids ^{are now well}

described [1]. ^The degree ^{of coherence induced} by ^the

very linear structure of polymer molecules is conve-

niently ^observed through two-body correlation func- tions characterized by screening lengths exhibiting specific concentration and chain length dependences.

These reflect scaling properties established from a

magnetic analog considered in the critical temperature range. However, descriptions of fluctuations in time have not yet reached the same level of _accuracy.

Collective effects were first introduced through ^normal

modes characterizing any chain motion; they ^{are not} questioned to-day [2, 3] ; ^{such an} approach ^{leads to a}

relation between a molecular characteristic time TR

and macroscopic viscoelastic parameters ^{such as the} zero-shear rate viscosity, ’10’ and the plateau ^modulus

G’ : ilo oc G’ TR [4]. ^Two of the main problems arising ^{from the} description of fluctuations in time in

polymeric liquids ^{are to} give’ a ^true theoretical esti- mate of TR ^{and to} propose ^a direct experimental approach ^{to its} measurements. It has been recognized

for some time that a quantitative description ^{of mole-}

cular mechanisms responsible ^{for low} frequency

viscoelastic properties ^of polymer ^{melts must be based}

on a clear observation of single chain diffusion _pro- cesses ; these must be governed by TR. ^{Such an} approach results from the reptation picture ^first

introduced by ^De Gennes, considering ^that long

chains move through fixed obstacles determining ^a

tube along ^{the chain contour :} only the ends of a

chain have a choice of direction when it moves among other chains [5]. ^The reptation ^{model is based on a}

defect propagation along ^{the chain; it was the first} precise molecular mechanism proposed ^{to describe a}

chain diffusion _process. The _average diameter of the tube is currently considered as being determined from the _average chain segment ^{of contour} length, Le, separating ^two successive coupling junctions [4]. ^To

renew its configuration ^{a chain must diffuse out of}

its tube; ^{such a} one-dimensional _process requires ^a

time 1"rep ^oc M 3 (M is the chain molecular weight) [5] ; correspondingly, ^the chain diffusion coefficient is

expressed ^{as : D} oc ^{R 2} >/Trep ^oc m-2 ; R 2 > ^is

the mean square end-separation ^{vector of a} chain. This

analysis ^also applies ^to entangled polymer ^chain

solutions where the self-diffusion coefficient is _pre- dicted to be proportional ^{to M - 2 C - 1.75} (C ^{is the} polymer concentration) [5]. Contrary ^{to the} descrip-

tion given ⁱⁿ earlier studies [5, 6], the relaxation _process of a tube is now considered as hardly contributing ^to

the complete change ^{of the} topological organization

of a chain [7, 8] ; ^{such a} negligible contribution only

characterizes chain molecular weights ^much higher

than the characteristic value Me ^{of onset of} entangle-

ments. The time required ^{to obtain a} complete ^rene-

wal of chain configuration ^is TR

_Lrep ^oc M3, ^{in a} polymer ^melt ( TR ^{OC M 3 C 1.5 in a} semi-dilute solu-

tion). ^{In addition to the} reptation process there is a

diffusion of what De Gennes considered as a defect gas along ^the chain; ^this corresponds ^{to a} wriggling

motion of the chain within a « tube » ; ^it is characte- rized by ^a relaxation spectrum ^{whose the} longest ^time

is proportional ^{to M 2} [5]. ^More recently, Greassley

and Edwards attempted ^{to establish an} universal law

relating ^the plateau ^modulus G’ ^to simple ^chain

parameters ^{such as the Kuhn} step length, ^the polymer density [9]. The search of additional universal dyna-

mical properties necessitates to perform ^{direct measu-}

rements at a molecular scale of TR ^while ilo and GN ^are

obtained from macroscopic viscoelastic measurements without _any difficulty.

Any accurate measurement of D should allow the determination of TR. Studies of chain self-diffusion in concentrated polymer solutions and in melts began

about thirty years ago by using radio-tracer tech-

niques [10]. ^{Later on, the NMR} technique ^{was used to}

estimate the diffusion coefficient in molten polyethy- lene ; ^{D was shown to} vary ^as M -5/3 [11-13J. Recently,

the translational diffusion coefficient D of a series of labelled linear polyethylene ^fractions (3 ⁶⁰⁰ Mw

23 000) ^{in a linear} polyethylene ^matrix (M,, -- ^{1.6 x} 105) ^was measured from infrared microdensity. ^The

M - 2 chain molecular weight dependence ^{of D was}

observed with a good accuracy; it gives ^the strongest evidence for reptation ^{in an} entangled polymer ^melt [14]. ^More recently, Rondelez et al. developed ^{a mea-} suring technique ^{called forced} Rayleigh light ^scat- tering ; it is based on the labelling ^{of a few} polymer

molecules with a photochromic probe; the space scale is created by illuminating ^the sample ^{with a} pulsed interference pattern; the time scale is obtained from the decay of the diffracted intensity ^{of a} reading

laser beam interacting ^with photoexcited molecules, only. Rondelez et al. showed that the self-diffusion coefficient _vary as M - 2 C-’-’ in semi-dilute solu-

tions ; this is in good agreement with the ratio of

( R’ > ^over TR [15]. ^This measuring technique ^was applied ^to semi-dilute solutions of polystyrene ^chains

of molecular weight Mw ^as high ^{as 7.5 x 105 and at}

concentrations lower than 0.2 g/g. The self-diffusion

coefficient, D, ^{has been} recently measured from a

pulsed ^field gradient ^NMR technique which also

applies ^to studies of semi-dilute solutions of entangled

chains ; the concentration and chain length depen-

dences of D were also shown to be in good agreement with scaling ^and reptation predictions [16]. Finally, ^it

must be mentioned that quasi-elastic neutron scat-

tering experiments ^{have been} recently applied ^{to stu-}

dies of single polymer ^chain dynamics in solution and the melt, considering short chains (MW 5 ^x 104)

and concentrations equal ^{to 3 x 10-2} g . cm- 3. ^Main

features of these studies were interpreted ^{in accor-}

dance with the Rouse-Zimm description [17]. However, the three above measuring techniques apply ^{to measu-}

rements of self-diffusion coefficients higher ^than 10-9 cm2 S- I ; they ^cannot be extended to studies of

polymer melts because D would be far too small for

long ^chains.

The present paper aims at showing ^{that NMR} might ^{be a direct} approach ^{to the} investigation ^of spectra ^of long chain relaxation times in polymer .melts ^without using any labelling ^or any external mechanical excitation. Time fluctuations of polymeric liquids around the thermodynamic equilibrium ^are

characterized not from measurements of chain-diffu- sion coefficients but from the observation of quantum coherence properties ^of magnetic ^{moments of} protons linked to strongly entangled chains. The principle ^of interpretation ^{of NMR} measurements is based ^on the two well established following properties :

i) ^The spin-system response observed from protons linked to very long polymer ^{molecules in a melt}

exhibits a solid-like character because of the presence of residual tensorial spin-coupling, ^6, equal ^{to about}

10~ 102 Hz. These are also observed on polymeric gels [18-25]. ^Tensorial spin-interactions ^{are not ave-} raged ^{to zero} because of an average orientational order of monomeric units necessarily ^induced by ^the

confinement of _any given chain within some volume

(a ^« tube », probably). The chain motion is perceived

as non-isotropic in the NMR time scale.

ii) ^A liquid-like response of the spin-system ^is

obtained at a given temperature by shortening polymer

molecules or by diluting ^{them. The} chain motion is then perceived ^as isotropic ^from NMR, although polymer ^{molecules are still} entangled with each other.

The observation of a transition from a solid-like behaviour of the spin-system ^{to a} liquid-like ^one necessarily ^{means that a} slow molecular diffusion is involved in the relaxation _process of the transverse

magnetic component. ^The longest relaxation ^time T’

characterizing this molecular _process must depend

upon chain molecular weight ^and polymer ^concen-

tration. The solid-like _response is characterized by 6T’ >> 1, ^{while the} liquid-like ^one is characterized by 6Ti « ^1. During the so-called motional averaging

process, TR ^must obey ^the approximate ^condition (5TR ^1. Therefore, Tv !-- ^{0.1 s} may be defined as the time interval required ^to perceive ^the chain relaxation process (or ^a part ^of it) ^{as an} isotropic motion. The criterion of isotropy ^{is based on a} measurement of 6;

6 _may be used as a reference frequency. ^{The relaxation}

time TR ^is probably shorter than TR. ^{One of the} problems arising ^{from NMR} studies is to identify ^the

chain motion (or ^a part ^of it) actually perceived ^from

NMR and corresponding ^to Ti rr ^{0.1 s. Two} attempts

to use NMR according ^to the above analysis ^have

been recently reported : they showed that the irrever- sible dynamics ^{of the} proton transverse magnetiza-

tion _may be sensitive to the _very slow diffusion _process of polyethylene [26] ^or polydimethylsiloxane (PDMS) [27] ^chains through topological constraints; T’ ^was

found to be equal ^{to 0.4} s, while Tv

tjoIGNO

^{1.2 s}

for a chain molecular weight M,

^{2.8 x 105. Several}

conditions are required ^{to obtain} non-ambiguous

NMR investigation ^{of the} dynamics ^of entangled polymer ^{chains :}

i) The relaxations must be grouped ^{into two well-}

defined dispersions Sll ^and Q2; Q, corresponds ^to

short times, it is called the transition spectrum; ^while Q2 corresponds ^to long times, it is called the terminal spectrum; ^it characterizes large ^scale rearrangements of chain conformation; TR ^{will be} supposed ^{to be the} longest relaxation time of the Q2 spectrum.

ii) Basic NMR properties ^must be related in a

simple way ^to elementary polymer properties specifi- cally ^induced by ^{the linear structure of a chain.}

iii) ^The spin-system ^must be defined in such a way that single ^chain magnetic properties ^{are observed} although all chains are in dynamical interactions with

one another.

iiii) To be used as a clock, ^residual spin interactions must have a right ^{order of} magnitude compared ^with

relaxation times of the Q2 spectrum; otherwise nuclei should be changed (13C, D...).

In the reference [5], ^{De Gennes} suggested ^that spin- dipolar couplings ^modulated by ^the reptation ^motion

of a chain should induce a characteristic mg ^{3l4 Larmor} frequency dependence ^{of the} spin-lattice ^relaxation

rate; ^{this has not been} yet observed. Considering ^also

the longitudinal component ^{of the} magnetization ^of protons ^{linked to} polyethylene chains, Kimmich et al.

recently ^measured spin-lattice relaxation rates at Larmor frequencies ^{as low as} 104 rad . s-’ ; ^results

were interpreted using several chain relaxation times characterized by ^{different molecular} weight depen-

dences [28].

A simple description of chain confinement effects will be given in section 2. The schematic approach ^to

the description ^{of chain} disentanglement effects is _pre- sented in section 3. Residual tensorial spin interactions

are calculated in section 4 using ^a freely jointed ^chain

model. Effects of a fast disentanglement process ^on the free induction decay ^{of the} magnetization ^{are calcu-}

lated in section 5 by using ^{a Rouse model to describe}

the diffusion _process of a chain. This model was chosen for the sake of simplicity although it is considered as

roughly describing viscoelastic properties ^observed

on a melt. The main reason why this model was used

is that it gives ^the possibility ^to exactly ^calculate NMR , properties instead of arbitrarily introducing empirical spectra ^as it is usual. No model has been proposed

until now to describe the transition from ^a solid-like behaviour of the spin-system ^{to a} liquid-like ^{one. The}

use of a Rouse model means that NMR is considered

as being ^{sensitive to the} disengagement process of a

chain from its tube more than to its wriggling ^motion.

In that case a reptation model should be considered,

but to our opinion ^{the Rouse model} gives ^{a reasonable} qualitative picture of variations of the relaxation process of the magnetization when the chain diffusion process is faster. Furthermore, ^{the Rouse model and} the « reptation » ^one have been shown to give ^similar

results for the mean square displacement ^{of a} point ^of

the so-called primitive chain and the time correlation function of its tangent ^vector. However, ^{the neutron} dynamic ^structure factors obtained from each model differ from each other [29].

2. Chain confinement effect : temporary average orientational order.

Considering ^a chain enclosed in its tunnel, any submolecule must be pictured ^as being

confined within a space volume where a field of tempo- rary constraints is exerted on every monomeric unit;

the submolecule is not at equilibrium ^{and its} entropy is reduced. Rotational isomerization processes of monomeric units of the free chain are now hindered by

constraints which might be described from an additio- nal potential energy function V(Rn, t) depending upon the position Rn of the nth monomer unit at time t. The confinement of a submolecule gives ^{rise to a end-} separation vector not averaged ^{to zero over a time}

scale smaller than TR. ^The determination of the temporary field of constraints is of course a very diffi- cult problem; ^{it is more} convenient to assume that _any submolecule out of equilibrium ^is fully characterized from its non-zero end-to-end vector reo Such an

approach ^{will be} conveniently illustrated from a

freely jointed ^{chain model;} this has been widely ^{used to}

establish first order properties specifically ^induced by ^{the very linear structure of} polymer molecules,

details about the rotational isomerization of mono-

Fig. 1.

^The freely jointed chain model. Bonds have equal length; ^every link carries a proton pair; Bo ^{is the} steady magnetic ^field.

meric units being ignored. Structural units are repre- sented by ^{beads of} equal mass, m ; they ^are freely

connected by ^{bonds of} equal length, a (Fig. 1). Any

submolecule is supposed ^to be built from Ne ^skeletal

bonds. The partial partition ^function corresponding

to a fixed end-to-end vector, re, is expressed ^{as :}

Possible orientations of _any bond aj ^{are described}

by the function f(aj); ^{the integral} is restricted to orientations of aj ^vectors compatible with the condition

with

and

the expression exp(- iq.a) may be considered as a parametric potential energy function perturbing

the connectivity ^function /(a) ^{of skeletal} bonds;

it reflects the _presence of a constraint exerted on a

submolecule and corresponding ^{to an end-to-end}

vector different from zero. It is the _way we shall describe chain confinement of a chain in its tunnel.

The probability distribution function of orientations of _any bond _a, given ^an end-to-end separation ^vector

re, has been already calculated; ^{details are} given

in several references [30, 31].

with a; ^{the mean} square end-to-end vector of _any submolecule. A temporary average orientational order of skeletal bonds is induced by ^chain elongation.

This property, ^of course, applies ^to real chains too;

the orientational order effect is probably ^enhanced by strong correlation of orientations of successive monomeric units. In the case where re a e formula (5)

is simply expressed ^{as :}

Skeletal bonds behave like molecules carrying ^a

permanent ^electric dipole ^and experiencing ^{an electric}

field.

Considering ^now real chains built from monomeric units carrying (-CH3-) methyl proton groups ^or

(-CH2-) methylene groups ^or (-CH= CH-)

methine _groups, it is clearly ^seen from formulae (5)

and (6), that internal proton dipolar interactions

JeD(t) established within such _groups are not averaged

to zero when the end-to-end vector re of _any real chain is different from zero : ( JeD(t) >re # ^{0. This is}

the fundamental property which will be discussed and applied ^{to the} observation of chain dynamics throughout ^the present paper; ( JeD(t) >r. ^{will be}

used as a low reference frequency ^to describe the well known NMR property called motional narrowing

effect [32].

3. Chain disentanglement ^{effects : a schematic} approach.

^{3.1 1 A} TWO-STEP AVERAGE.

It is well known that magnetic relaxation rates are expressed

as Fourier transforms of the correlation function, 0(t), of tensorial spin couplings, JCD :

We consider that as a consequence of the existence of two sets of chain relaxation times, ^the averaging

process of tensorial spin interactions must occur

in two steps [33]. The correlation function 0(t) ^is expected ^{to exhibit two} characteristic time ranges;

the first one corresponds ^to short relaxation times of the Q, ^set; the second one is associated with long

range changes ^{of chain} configurations, characterized

by ^the Q2 ^set. Considering ^{as a first} approximation,

that _any submolecule out of equilibrium may be defined from its end-to-end vector, r., the maximum

amplitude ^of 0(t) ^{is written as :}

where >re = y ^{ },,, ^{means a summation over}

all chain configurations compatible with the end-to- end vector re. The partial average may also be _expres- sed as :

The first term of the right ^hand part of the above

equation ^{is related to local} fast motions occurring

within the submolecule defined by re ; effects of these random motions on JCD(T) ^are described from a corre-

lation function r r e (t) reflecting properties ^{of the set}

of chain relaxation times Q, :

all chain segments confined within a given submolecule

are supposed ^to undergo ^a relaxation process ^towards the partial equilibrium ^defined by re. 0(t), ^{is a} rapidly decreasing ^function towards JeD(t) );e’ ^{over a time}

scale about equal ^{to 10-5 10-7 s.} Furthermore,

submolecules are involved in long range configu-

rational changes ^of polymer molecules associated with the tube renewal _process; this is governed by

chain disentanglement; accordingly, the end-to-end vector is actually ^a time function r,,(t) and effects of its random variations ^on JeD(t) ^are described from

a correlation function yre(t) ^{with :}

T,,.(t) ^{closely reflects} properties of the second set of chain relaxation times Q2 ; it decreases towards zero over a time scale about equal ^{to 10-1 s. The corre-}

lation function 0(t) ^{can be} splitted ^{into two} indepen-

dent parts ^when Qi ^and S22 ^{are well} separated ^from

each other; local motions and submolecule diffusion

being considered as stochastically independent pro-

cesses (Fig. 2).

Nuclear magnetic relaxation rates are known to be

roughly expressed ^{as :}

for the spin-lattice relaxation rate; mo is the Larmor

frequency ^{of the} spin system : ^{mo = 2n} x107 rad . s-1;

yre(t) ^cannot contribute to the expression ^of Tï1

because its characteristic times are too long compared

with (t)o 1. ^Also,

for the spin-spin relaxation rate ; ^the contribution of rre(t) ^to (T2"l )re ^{is now} negligible because it decreases too rapidly compared ^with y,,,,. ,(t). ^{An average}

over all end-to-end vectors should, ^of course, be

Fig. ^2.

The correlation function of tensorial spin-couplings

is supposed ^{to exhibit two} well-characterized time variation ranges. The first ^one reflects properties of the transition spectrum Q, ; the second one is associated with large space scale rearrangements of chain conformations : it reflects the

Q2 spectrum; ,.(O) = JCD(T) >2 r.-

considered. The above qualitative approach clearly

shows that two mechanisms _may be responsible

for magnetic relaxation _processes of protons ^linked

to entangled long chains. The above schematic

approach ^{will now be} applied ^{to some} specific ^cases.

High frequency motions within a given submolecule will not be considered in the present paper.

3.2 STRONGLY ENTANGLED CHAINS.

It is now

supposed that JeD(t) >.:: 1 is shorter than all chain relaxation times of the Q2 ^{set, because} protons ^are linked to very long ^chains; the renewal time TR ^{of the}

chain configuration ^is supposed ^to obey ^the inequality:

Accordingly, the well known NMR property ^called motional narrowing effect is only partial. ^Tensorial

interactions are not averaged ^{to zero} during ^NMR

measurements : there is a residual spin interaction

( JeD(t) B. and the motion of the whole chain is

perceived ^as non-isotropic although ^it actually ^has

a spherical symmetry ^{when it is observed on a time} scale long enough; the second characteristic time range of the correlation function 0(t) ^{is not observed}

from NMR because it lasts too long. ^{A solid-like}

response of the spin-system ^is expected ^{to be} obtained;

it is governed by JeD(t) re ^{and it may} be characte- rized like any response which would be obtained

from I JCD I ⁱⁿ ordinary ^solid spin systems. However,

I JeD(t) I ^is considerably ^reduced by local motions :

JeD(t) >,e, I -10-3 ^{1 JCD 1. A spectrum narrowing}

effect _may be conveniently ^induced by rotation of

samples ^at angular frequencies higher ^than

I ^{; JeD(t)} >re I ^{~ 2 7r x102 rad. S-l. Such an effect}

is easily ^observed using conventional high ^resolution spectrometers [34]. Also, ^a pseudo-solid spin-echo

may be obtained using appropriate pulse

sequences [21]. ^{All these} properties ^{have been} widely

studied on different polymer systems [22-24].

When polymer ^{molecules are} strongly entangled,

an average orientational order of monomeric units is induced by ^chain confinement; ^{it is} easily ^observed through JeD(t) re, ^from a resonance spectrum ^or

a free induction decay. Analogous properties apply

to polymer network structures, ^too.

3.3 CHAIN DISENTANGLEMENT EFFECT : : NUCLEAR MAGNETIC CLOCK.

It is now considered that the renewal time TR ^{is set} slightly shorter than the inverse of the residual spin interaction ( JeD(t) re 1 ^{by shorten-}

ing polymer ^molecules ( TR ^OC M3) ^or by adding

solvents ( TR ^{OC M3} C) :

When TR and C JeD(t) >r ’ obey the above ine-

quality, the second step ^of the ^motional _narrowing

effect is achieved. The full motional averaging ^of X,(t) may be observed from NMR; ^{it must reflect} properties ^{of the} Q2 ^set of chain relaxation times

(Fig. 2). ^{This is a} way ^to investigate long range chain

dynamics ^{effects. It} clearly appears that the characte- ristic chain correlation time measured from NMR is the time interval necessary ^to observe the motion of a chole chain as isotropic; the observation of

( JCD(T) >,,,, ^{is used as a} criterion of non-isotropic property. The reference frequency ( JCD(T) re strongly depends upon : i) ^{the nature} of observed nuclei

(1 H, D, 13C) ; ii) the spin interaction : dipolar ^or quadrupolar; iii) the chemical structure of monomeric units.

Observations of an average orientational order of monomeric units or of a disentanglement ^effect

on NMR may be applied ^{to the} study ^{of classical} thermodynamic phenomena ^{such as a} partial polymer crystallization process ^{or a} chain demixion _process.

3.4 PARTIAL POLYMER CRYSTALLIZATION.

Consi-

dering strongly entangled polymer molecules, ^their partial crystallization ^is expected ^{to have at least}

two effects on the residual tensorial spin coupling JeO(t»re. ^It first enhances chain confinement because most polymer ^{chains are} partly ^or entirely ^embedded

in crystallites ; then, ^the presence of platelets ^{in solution}

in the liquid part ^{of the} polymer system may slow down the chain diffusion _process by increasing ^friction

coefficients. The 02 spectrum ^is probably ^shifted

towards longer chain relaxation times. _{In any} case

the strength of the residual spin coupling is increased

as a function of the crystallinity degree. ^{The kinetics}

of partial crystallization ^of cis-1,4-polybutadiene

chains has been actually followed from NMR measure- ments over a time scale ( ¹ min.) usually ^considered

as uneasy when dilatometric measurements are

performed [35].

3.5 CHAIN DEMIXION.

The effect of chain disen-

tanglement ^{on NMR} properties may be used to observe chain demixion _processes on a microscopic space

scale; starting from concentrated homogeneous ^chain

solutions characterized by ^a given ^residual spin

interaction a chain demixion _process can be induced

by lowering ^the temperature ^{of the} polymer-solvent

system ^or decreasing its chain concentration. Consider-

ing ^the polyisobutylene-toluene system, ^a microphase separation has been shown to occur; it is characte- rized by the absence of _any residual interaction. A

phase separation ^curve has been drawn by observing

the strength of the residual interaction as the polymer

concentration is decreased or as the temperature ^is lowered; ^{this curve was shown to} obey ^the equation :

C oc (0’ - T), ^{with 0’}

^{265 K} while the theta- temperature ^{is known to be 260 K} for infinite chain

length [36]. ^{This curve} is in accordance with the phase diagram proposed by ^Daoud and Jannink [37].

However, ^no phase separation ^{was observed at a}

macroscopic space scale; therefore, ^{it is believed}

that a microphase is defined in the system; ^{it consists}

of collapsed coils smaller in size than a whole polymer

chain; ^{such coils are} binded to one another by long

chain segments surrounded by solvent. These long

filaments should correspond ^{to the low} polymer

concentration phase; ^the binding ^{is loose} enough

to induce a motional averaging of ( JeD(t) )re.

4. Residual tensorial spin interactions.

41 THE

SPIN-SYSTEM MODEL.

Using ^a freely jointed ^chain model, every link is supposed ^to carry ^a proton pair;

the distance between two nuclei located on a same

link is b ; ^all magnetic interactions between different

pairs ^are neglected. The relaxation function of the transverse magnetization ^of protons ^{bound to such}

a chain will be calculated within the adiabatic approxi- mation, neglecting contributions induced by ^fast

isomerization _processes of monomeric units. The

spin-system ^{is a} two-spin system, taking dipolar

interactions within each pair into consideration.

4.2 THE GLASSY STATE.

It is convenient to start the description ^{of NMR} properties ^{from the state}

in which chain segments ^{have no} large amplitude

motions at all : the glassy ^state. They only undergo high frequency vibrations of small amplitude ^which

cannot govern the spin-spin relaxation process. Pro-

perties ^{of the} spin-system associated with a proton pair ^{are well known; its} hamiltonian is :

with

,Xz ^{is the} Zeeman energy and,

with A12

T 2 h (3 COS2 0, _ 1)/2 b 3 ; y is the _gyro-

magnetic ^{ratio of a} proton; ea ^{is the} angle ^{which the}

bond a makes with the steady magnetic ^field direction, Bo.

Je has four eigen functions I ^{+ +} X (I ^{+ - ) +}

! - + >)/,/2-, 1 - - > and (I + - > ⁺ >)/,,/2-,

corresponding ^to eigen ^values E+ = - (t)o, Eo = 0,

E_ = Wo and Eo

^{0. The} stationary ^state (!+-)-!- ⁺ >) /,/2- ^{is not} involved in the

description ^{of the} dynamics ^{of the} transverse compo- nent of the magnetization A.,. The normalized relaxation function expressed ^{in the} rotating ^frame

is :

with

where C )orient. ^{means an} average ^over all bond orien- tations in the glassy state ; eo (a)

³ T’ h (3 cos’ 0,, - 1)/

4 b3. The resonance-line spectrum is the Fourier transform of GG(t) ; it has been widely described;

each proton pair gives ^a doublet; ^its splitting depends

upon bond orientations [38, 39]. The second moment of the spectrum ^{is :}

with

4.3 RESIDUAL SPIN INTERACTIONS.

Strongly entangled ^{chains are now} considered assuming ^that

the two sets of chain relaxations Ql 1 and 02 ^{are well} separated from each other. The relaxation function

expressed ^{in the} rotating ^{frame is now} expressed ^{as :}

This function is conveniently calculated using ^a

cumulant expansion ^{to the second} order and consi-

dering ^a given submolecule characterized by ^{its end-} separation ^vector r,, :

with

and

T(r,,) ^{is the} probability distribution function of end-to- end vectors re ; qJre(t) is ^a relaxation function induced

by fast rotational isomerization of skeletal bonds

keeping the end-to-end distance as fixed; qJre(t) ^has

the mathematical structure of a spin-lattice ^relaxation function; the order of magnitude of its time constant should be that of a spin-lattice relaxation time T, (Tl ;$ ^{1 s: observed on} cis-1,4-polybutadiene (PB), polyisobutylene (PIB) ^and polydimethylsiloxane (PDMS) ^for example). Contrasted ^to qJre(t), ^{the term}

cos « 80 t) has the mathematical structure of a broa-

dening mechanism observed on solids:( go

10- 3 s. Accordingly, (Pr,,(t) ^{has a} negligible ^contribu-

tion to the time variation of Gre ^{(t) ;} it will be considered

as an additional small broadening of the main spectrum determined by 80 )re.

The residual spin interaction can be calculated from

the probability distribution function of orientations

of a bond a given by ^formula (5) ; details of such a calcu-

lation have been given in several references [30, 31] :

with

and x

3 re/Ne ^{a ;} ere ^{is the} angle which the end-to- end vector of the submolecule makes with the steady magnetic ^field Bo. ^The limit value of L*(x) ^{when x is} large is, ^of course, ^{one :} all bonds are oriented along re.

At small x values L*(x) ^ x2/15. ^{The width} of L*(x)

is about N, ^a, while the width of the distribution func- tion of re is-.,/Ne a; accordingly, only small values of x

are involved in calculations :

The above formula is a bridge connecting ^{low fre-}

quency NMR properties ^{to chain} dynamics ⁱⁿ poly-

meric liquids. It may be applied in several _ways to the

study of polymer system problems involving ^a two-step motional averaging ^of spin dipolar (or quadrupolar) couplings.

4.4 VALIDITY TEST : POLYMERIC GELS.

Recent NMR studies of polydimethylsiloxane gels ^obtained by end-linking reaction have been considered as a test of

validity of formula (29), although ^{it has been esta-}

blished from a freely jointed chain model [31]. ^{It was}

inferred that formula (29) ^also applies ^to real chains except ^{for some} numerical factors. The relaxation function of protons ^{of one} methyl group attached to a

PDMS chain is well appropriate ^{to the} study ^of

formula (29) because it is expressed ^{as :}

where c is the three-fold axis of the (-CH3) group. The above simple formula results from an average effect of

dipolar couplings between the three nuclei, ^induced by

the random rotational diffusion of the methyl group around the c-axis; ^this high frequency process is

supposed ^{to be} involved in the actual relaxation function through ^an additional small broadening ^of

the main spectrum determined from G H, (t). ^Details

about the _energy spectrum ^{of a} three-spin system,

including dipolar couplings, have been widely ^dis-

cussed in several references ; it ^{is not the purpose of the}

present paper ^to give them; ^{it is} only considered that the relaxation function G H3 (t ) depends ^{on the random}

diffusion of the c-axis tightly ^{bound to the chain}

skeleton [27, 31, 38, 39]. Accordingly, GHI(T) may

reflect a residual dipolar energy induced by ^{a chain}

constraint such as a confinement. There are at least two questions arising ^{from the} description ^{of the} swelling process of polymeric gels [1]. ^{The first one}

concerns the so-called packing condition of partly

swollen elementary ^{chains. The second} one concerns

the role of crosslink points considered as defining

end-to-end vectors, _re, of elementary chains. The relaxation function was observed from a spin-echo experimental procedure ^on tetrafunctional gels ^built

from calibrated elementary ^{chains. It was} clearly

shown that the relaxation rate is a function of the

swelling ratio, q, of the gel according ^{to the law :} q’13

(q is the ratio of the partly ^swollen gel ^{over the volume}

of the dry gel). ^{Such a} result associated with gels ^cha-

racterized by ^different elementary ^chain lengths ^was

considered as a test of both relations :

reflecting ^the packing condition of elementary chains;

N, is the number of skeletal bonds in a given ^elemen- tary chain. The role of crosslink points ^{was investi-} gated by measuring the relaxation rate (T2"l)* ^{of the}

transverse component ^{of the} proton magnetization ^of gels ^{swollen at} equilibrium. ^From the c*-theorem

proposed by ^{De Gennes, the} swelling ^ratio q* ^at equili-

brium is known to vary ^as q* ^oc N:/5 ^{as an} elementary

chain length ^function [1]. The relaxation rate (T21)*

measured at equilibrium swelling ^{was shown to} syste- matically obey ^{the law :} (T2 q*

^{constant num-} ber, whatever the chain length (200 N,, 480).

Such a result is a test of the dependence of the residual interaction _upon elementary ^chain length variations :

At equilibrium swelling r* ^oc N’I’ ^and (r*’)IN’ ^oc Ne-4/5 ^oc (q*)-’. ^As it will be shown in the next section, the Ne 2 dependence ^{of the} magnetic relaxation rate

indicates that large amplitude fluctuations in _space of crosslink points ^{are not} perceived from NMR. All end-to-end vectors of elementary ^{chains are considered}

as having ^{about the same} length I re I but different orientations of _course; from the reference [31]

p is the density of the undiluted polymer, A ^is Avoga-

dro’s number; Mp is the monomeric unit molecular

weight ^{and a is a factor} taking ^chain linking into consi- deration. The magnetic relaxation function is written

as :

The good ^{agreement of} experimental results with the above formula suggests ^{to extend it to stretched}

polymeric gels :

The direction of stretching ^is supposed ^{to be} parallel

to the steady magnetic field direction Bo ; A ^{is the}

extension ratio. When the stretching direction is not

parallel ^to Bo, ^{the above formula must} be transformed like second order spherical harmonics. The effect of

stretching ^of crosslinked rubber has been unambi-

guously ^{observed on} molecular orientation of solvent from quadrupolar relaxation [40]. ^A numerical illus- tration of the G13(t) function is reported ^on figure ^3.

Numerical values of the model parameters areAG

A-’ X- 1)213 = 1. ^No additional broadening ^contri-

bution has been introduced to clearly observe varia- tions of the signal modulation induced by gel ^stretch- ing.

Note that the property : q*(T2 ^{= const., has}

been called C*-theorem [31].

4. 5 STRONGLY ENTANGLED CHAINS.

4. 5 .1 Ideal chains.

The description ^{of NMR} properties ^observ-

ed on polymeric gels is contrasted to that of properties

observed on entangled chains because an average ^over all lengths ^of end-to-end vectors of subchains must now

Fig. ^3.

Free induction decays calculated according ^to

formula (33); values of 2 G13(t) -I ^are actually reported.

Numerical values are : Jc

104 rad.s-’ ;

swelling ^{ratio q}

4; stretching ^{ratio :} A : A = 1 ; B : A = 2;

C : h

4. Such decays could illustrate properties ^{of a} poly-

meric gel ^with elementary ^chains carrying methyl groups

(PDMS, ^for example). ^No additional relaxation mechanism induced by spin-couplings ^{between different} methyl ^groups

has been introduced. The time scale is 10.24 ms.

be considered. In other words, ^the magnetic ^relaxation

function associated with a polymeric gel resembles the

powder average of the well known Pake doublet; ^the

average ^over crystallites orientations is replaced ^with

an average ^over orientations of end-to-end vectors ; while in entangled ^chain systems, orientations and also distances have to be averaged. ^This average has been

recently calculated starting ^{from a} probability ^distri-

bution of end-to-end vectors, T(r,,), equal ^{to :}

with Qe ⁼ Ne a2 ; Ne is the number of skeletal bonds

defining any submolecule. The relaxation function

G 112 (t) ^{has been} calculated in the reference [41] ^consi- dering proton pairs ^or methyl ^{groups :}

with

and

The second moment of the resonance spectrum observed in the molten state is :

For a freely jointed ^{chain :}

Formula (39) illustrates effects of chain length ^and

of chain flexibility ^{on NMR} properties, although ^{it is}

calculated from a freely jointed ^{chain model. As it}

may be expected, ^{resonance lines are broadened} by increasing Jfl (lost ^{of chain} flexibility) ^or by decreasing Le (lost ^of entropy). The reference time scale of G H2 (t ) is b -1, ^{with 6 =} J02 ^Coo dG/Ne ; it may be expressed, using ^a reduced variable v

6t; G 12 (t) behaves like

a v -3/2 function, ^at large ^time values; ^{as it is} expected GH2(t) differs from a pure exponential ^{time function;}

a numerical illustration of G 12 (t) ^{has been} reported

in reference [41] ; its Fourier transform looks analogous

with the so-called super-Lorentzian ^{resonance lines}

observed on many molten polymer systems.

4. 5 .2 Real chains.

Although the relaxation func- tion G "2 (t) has been established from a freely jointed

chain model, it has been adjusted within less than 5 %

of uncertainty ^{to NMR} spectra ^{observed on} strongly

entangled ^real chains such as cis-1,4-polybutadiene

molecules, considering the submolecule length Ne ^{as a}