Rheological Rouse model for a polymer in a nematic matrix

HAL Id: jpa-00212556

https://hal.archives-ouvertes.fr/jpa-00212556

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Rheological Rouse model for a polymer in a nematic matrix

F. Lequeux, R. Hocquart

To cite this version:

F. Lequeux, R. Hocquart. Rheological Rouse model for a polymer in a nematic matrix. Journal de

Physique, 1990, 51 (22), pp.2595-2604. �10.1051/jphys:0199000510220259500�. �jpa-00212556�

2595

Rheological Rouse model for a polymer ^{in a} nematic matrix

F. Lequeux (1) ^{and R.} Hocquart (2)

(1) I.C.S.-E.A.H.P. (C.N.R.S.), ^{4 rue} Boussingault, ⁶⁷⁰⁰⁰ Strasbourg, ^France

(2) ^{Lab. de} Spectro. ^et d’Imag. ^Ultras. U.L.P., Institut Le Bel, ^{4 rue Blaise} Pascal, ⁶⁷⁰⁷⁰ Strasbourg ^{Cedex, France}

(Received ¹⁸ May 1990, revised 13 July 1990, accepted ²⁰ July 1990)

Abstract. 2014 We present ^and develop ^{a model of} coupling ^{between a Rouse chain and a nematic}

order parameter. ^The geometry ^{of the} chain, ^{for each} mode, ⁱⁿ simple shear flow is calculated.

We discuss local torque and normal stress sign. ^This implies ^non trivial ways ^to introduce the

opposite coupling (from ^{chain to} parameter order) ^{in order to} get ^a model for Rouse nematic

polymers.

J. Phys. ^{France 51} (1990) ^{2595-2604 15 NOVEMBRE} 1990,

Classification

Physics ^Abstracts

61.30 62.10 - 61.40K

Introduction.

Rheological observations on nematic polymers melts show up ^some peculiar properties ^of

these systems. One of the most surprising ^{is the first normal stress which can be} negative ^for

some range of shear ^rates. At the present time, ^{there is no} relevant model for these polymers.

At large ^scales (compared ^{to the} persistence length), ^the polymer ^{behaves like an} anisotropic

Rouse chain [1], ^{but at short scale} (persistence length), ^{it is a} nematic, ^{with an order} parameter [2].

There is some influence both from the Rouse chain to the order parameter, and from the order parameter ^{to the chain} geometry. ^{Hence a} complete ^{model must take the two} couplings

into account : the first one describes the geometry of the chain in a flow with a given ^nematic

order parameter, ^{while the} second describes the order parameter ^{in a flow with a} given polymer geometry.

In this _paper, we are only concerned with the first coupling ^{in dilute} systems. ^{We then} study

the behavior of dilute chains with an anisotropic elasticity, coming ^{from the order} parameter.

It is in fact a generalized ^« Brochard » model of polymer solutions in nematic liquids [3]. ^This

model represents ^the first order coupling ^{between a nematic and a} polymer ^{in a flow. It must}

at least be coupled ^{with a second} coupling ^{in order to} get ^a nematic-chain model.

At last, the flow is a simple shear flow and the nematic director is contained in the plane ^v, Vv, ^{but the} formalism described here could easily be extended to other geometries.

1. Description ^{of the} coupling ^model.

Let n be the nematic director (n ^is equivalent ^{to -} n).

The chain is considered as an elastic spring : - r(s) ^{is the} position ^{of the chain.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510220259500

s being the coordinate which represents ^{the monomer} number, ^as usual convention for continuous Rouse model (0 -- s «-- L).

kl ^and k2 ^{are two elastic constants of the} spring, ^{the first} being the classical constant, while the second represents ^the anisotropic coupling ^{constant which} is, ^of course, ^a function of the order parameter, (say S), ^{such that} k2(S ⁼ 0) ^{= 0.}

We choose a generalisation ^{of the «} Brochard » free energy (see [3]) :

One can easily ^{see that} k2 ^{must be} negative ^{in order to describe a} main chain nematic polymer (where ^{the chain tends to} align with the director [4]) ^{and the} stability ^{of the chain involves} - ki «-- k2-

This energy gives ^{the force} along ^{the chain as a function of} its extension and its orientation.

Minimizing the free energy, ^we get ^{the force F in monomer units :}

In the same way, ^{one can} deduce the torque ^{F on} the nematic which is equal ^{to :}

2. Langevin équation ^{of the chain in a} simple ^{shear flow with n constant.}

There are in general ^two equivalent ways in order ^to write complete statistics of an object

submitted to a hydrodynamical ^{movement. The classical} way is to write the Smoluchovski

equation ^derived from the usual continuity equation (i.e. the fluxes in the phase space). ^The

second is to write the equation of motion of a Brownian object. ^{In our case, the latter} equation, ^a Langevin equation ^{is a} very suitable method.

Let D be the flow rate tensor, and C the friction coefficient of the chain by ^monomer.

The general equation ^{is then} [5] :

where A is a random force acting ^{on the monomers with the} following properties :

In fact A2ij ^{must only have the symmetry of ni} nj ^{but we choose it} proportional ^{to the} unity

tensor as a hypothesis ^{of our model :}

At equilibrium, this choice does not change ^the physics ^{of the model, because the} changes

in Aij ^{and in ki are} equivalent. ^{In a} flow, ^the equivalence ^is lost because of the non-compatible symmetry ^{between D and nn. So we} limit ourselves to an isotropic ^{random force, as well as to}

an isotropic ^friction coefficient C (on ^the contrary ^to F. Brochard who uses a anisotropic

friction coefficient).

In a simple ^shear flow, ^{D has} only ^{one non} vanishing ^{term :} Dxy ^{= g where g is the shear}

rate.

2597

The director being contained in the xOy plane, ^{as was} expressly stated in the introduction of this _paper, with n ⁼ (cos lp, sin _lp, 0 ), (See Fig. 1), ^{we have to} solve the three movement

equations :

’

Fig. ^1. - Geometry ^{of the} problem. cp is the angle ^of the director with the axe Ox. (J n ^{is the} angle ^{of the} anisotropy ^{of the n} Rouse mode coil. v is the flow velocity.

3. Solution of the equations.

A standard _way of treating ^{such a} system ^of equations ^{is to use} the Fourier transform

r (n, v ) ^{of the vector} r (s, t ), ^{which is} given by :

and the inverse relation

where we choose rG ⁼ 0, rG being ^{the center of mass} of the chain.

The boundary conditions are here implicit. ^In particular, ^we observe that

agreement with the conditions at both ends of the chain.

In the same way, ^A is transformed into A, ^so the three equations ^{become :}

where v is the frequency ^{and n} the number of the s-mode.

On the other hand, ^relation (5) is also transformed into :

If we consider the movement equations, we can see that they ^are linear and that the first two are coupled. ^{The solution} is then trivial : let D be the determinant of the first two equations :

where, ^for convenience, ^{we have} put :

Then :

We can see that z has neither dependence in g ^{nor in n} (Note ^{that n is in the} xOy plane).

4. Statistics of the chain at rest.

At rest (g = 0), ^{we can show that the} polymer ^{is no} longer isotropic, ^{but has a} spheroïdal

form. Choosing cp ⁼ 0, ^{we obtain :}

and

It is then easy ^to deduce

2599

For instance, the first average value is calculated according ^{to the relation :}

Hence :

In the same way, ^we get :

We have used the relation in order to obtain the numerical constant in the above relation.

We observe. that the polymer ^{coil is a} prolate spheroïd, its axis of revolution being parallel

to the director. The condition on k2 ( - k 1 -- k2 ) appears clearly ^{and the} sign of k2 gives ^the sign

of the anisotropy of the coil.

5. Calculation of the coil geometry ^{in a} stationary ^{flow with a} given orientation of ^n.

The calculation is only ^a generalisation ^{of the} previous ^{one and lead to :}

Which are the four non vanishing components of the second moment of the coil

Mij = C (ri - ^{rGi )} (rj - rGj ) %

The existence of such an integral ^{is not} trivial and we are going ^to show that for some values

of g, W 1, W 2 and ’P, the second moment M can diverge. ^{As a matter of fact,}

D ( v ) D ( - v) ⁼ DD * derived from (9) ^{is :}

and this quadratic ^{form in v 2 can have a} pole ^{for some values of} g. DD * ^can be written ^as follows :

So the integral

and diverges ^{for :}

if, ^{of course} N =F 0.

In the case where g is smaller than its values at the pole, ab > 0, (this corresponds ^{to the} only physical value of ab as explained further). ^We also have :

Equation (14) ^{means that} complete uncoiling ^occurs when g ^increases at ço constant. The

physical origin ^{of this} uncoiling ^{is the} amplification of the natural extension and of the

anisotropic diffusion due to the elastic tension on the chain. A similar extension _appears in extensional flow. Here the simple shear flow is compensated by ^the torque ^{due to the} anisotropic system. ^We will show further that spontaneous uncoiling ^{in this} system ^would require special coupling between the coil and the nematic.

In the next section, ^we will describe the geometry of the chain for given values of the shear flow and the angle ^ço of the nematic.

6. Geometry of the chain in shear flow.

The tensor M;j describes the second moment of the chain. It is in fact the sum of the contribution of each Rouse mode Mij. ^{Let us} discuss the anisotropy ^{and the} angle ^{of each}

Rouse mode.

6.1 ANGLE OF A ROUSE MODE. - The angle 03B8n, for which the extension is maximum in the

plane xOy, i.e. the maximum of (.X cos (J n ^{+ sin} (J n)2) ^{is given for :}

Note that MlZ ^{does not} depend ^{on the flow.}

For the isotropic ^{case, we} have the classical result [6] :

For g ^{= 0 we have of course} tg 2 On ⁼ tg ^{2 cp.}

If the interaction between M and the director n is _very strong, ^then 0 n = cp, and we get ^the following equation between g ^and en :

It is interesting ^{to look at} the situation when the shear ^rate is near 0.

2601

In this limit we get the relation :

which is n independent, therefore the total moment M has also this orientation. The angle ^of

the distribution 0 is no longer equal ^to the classical value (for isotropic systems) 77-/4 ^{for W2 =1= O. The strong} coupling between the director and the polymer ^{leads to the} following property : ^{the slow rate} limit of the distribution anisotropic angle ^{for our} system ^is

never the one of an isotropic system. ^In figure ^{2 is} plotted ^the angle of the chain distribution

as a function of the shear rate for several values of cp.

We will now show that the anisotropy ^can change in direction but not in sign.

Fig. ^2.

^Plot of 03B8n, ^{as a function} of g for several values of _’P. The values of ço reads by 03B8n(g = 0) = ço. The angles ^are given modulo 180°. There is a singularity ^for (cp ^{= -} 45 °, g ^{= £0} 2 ) ^and

we give ^{the two limits} cp -+ ’ ( - 45 ° ) (the ^lowest curve) ^and cp -+ - (- 45 ° ) ^{at the} top (cp ⁼ 13 5 ° ) . ^Values

of the parameters : w 1 = 1, W 2 = - 1/2, g in ^w 1 unit.

6.2. MAGNITUDE OF THE COIL ANISOTROPY. - Let us consider the two dimension moment

The anisotropy ^is given by the ratio of the two positive eigenvalues À and ^À 2. With the aim of

a simplification, ^{we will} just calculate the _square of the difference a between the two

eigenvalues (5 == Ik 1 - ’k 2) -

Let S and P be the sum and the product ^{of the} eigenvalues. ^{One has} easily the relations :

which, ^from (12) ^and (13), ^{leads to :}

The coil is isotropic only ^{for the two sets} (g ⁼ lJ) 2, ’P ^{- 7r} /4) ^{and (g = -} lJ) 2’ ’P ^{= - ’TT} /4).

In the plane (g, ^rp ), the coil is isotropic ^for only ^two points. & 2 ^is continuous in this plane ^and

vanishes only ^{for two} points, ^{hence 5} keeps always ^{the same} sign : ^{there is no} inversion of the coil while varying g ^{or e.}

7. Torque ^on the nematic.

As explained previously, ^the torque a-XY - Uyx ⁼ (yFx - xF y) ^is given by equation (3) ^which

reduces in our case to its z component :

In terms of Fourier components, rz ^{becomes :}

and from (8) ^and (10), ^{we find :}

Hence :

or

Finally, ^{after summation we} get :

where

and

Neglecting the solvant nematohydrodynamic torque, ^{we obtain the} equilibrium configuration

for a vanishing torque, ^{hence for T = 0.}

We will show that, ^{in this} case, for a simple ^shear flow, uncoiling can never occur. To prove this statement, it ^is enough ^to consider the n contribution to the torque. ^If we are near a pole

of DD*, ^{we can see that the} torque ^{will not} diverge only ^if (see (20)) :

2603

On the other hand, uncoiling is reached only ^if (15) is satisfied. This spontaneous uncoiling

in the shear flow requires ^both (15) ^and (22) ^to be satisfied. This gives ^the condition :

For physical ^reasons, this relation is never verified. (- (ù 1 : (ù 2 and w 1 ::> 0 ).

If we do not neglect ^nematic torque contribution, ^{the total} torque ^{must not} diverge, ^and uncoiling ^exists only ^{if both} the nematic torque ^{and the coil} torque diverge. Even in this _case, if there is a strong coupling ^{between the order} parameter ^{and the} coil, ^{as in} (18), ^with

0 ⁼ _e, because (18) ^and (15) ^{are also} incompatible, uncoiling ^cannot appear. It is then clear

that, except for very special ^{and non} physical coupling, the chain would not uncoil. This

implies ^that expression (15) ^is always positive.

8. Normal stress.

The first normal stress measured in semidilute nematic polymers ^{is found to be} negative ^for

some range of shear rates [7]. Qualitative explanations have been proposed [8], ^{but no}

realistic interpretations ^{have been} given ^{in the case} of flexible nematic polymers. ^Marucci proposed ^a convincing explanation ^{in the case of} rigid ^{rod nematic} polymers [9]. ^{It is then} interesting ^{to look at the} sign of the first normal stress difference given by ^our coupling, ^which

indicates what the zero order term would be, ^{and how chain} geometry ^{act on the stress.}

The first normal stress difference is given by :

According ^{to the} preceding calculation, ^{we obtain :}

After a straightforward integration, ^we get :

The denominator of this expression ^is always positive (see ^{in 5 and} 6) ^{while the numerator}

has two contributions. The first, proportional ^to g2, is the classical contribution : it is always positive ^{and does not vanish for} (JJ 2 ⁼ 0. The second represents ^a specific anisotropic

contribution : it has the opposite sign ^of cv 2 sin 2 _cp. In such ^a model, ^{where there is} only ^a polymer contribution to the stress, ^a negative ^{normal force occurs} only ^at small shear rate and if _{(JJ 2} sin 2 cp ^is positive.

In the usual case (where (JJ2 is negative), ^a negative ^normal stress cannot be described with a

classical nematohydrodynamic ^model. Using classical linear nematohydrodynamic, ^{one sees}

that the angle cp is given by the ratio of two viscosity coefficients a 2/ a 3 ^{= tg2 cp [10]. If this}

ratio is negative, ^{no stable} position ^{of the director can} be found. But if the ratio is positive,

stable value of cp is contained between 0 and ’TT /2. ^{In a linear model, sin 2 cp is} positive ^or

there is no stable position ^for cp, hence first normal stress difference is always positive ^for stationary ^flow.

But non linear models, in which the order parameter plays ^a great role, ^{as in} [9], ^{can answer}

the problem. ^{Such a} model, which would take the rigidity ^of polymer ^{into account, could}

induce ^a negative ^normal stress, ^{because a stable} negative ^{sin 2} ço structure could be found.

At last, ^the complete ^reduced expression for the normal stress :

gives, after summation on n :

where L and x ^are the same as in (21).

(Note ^{that this} expression ^{does not} diverge ^{for sin 2 rp =} 0).

Conclusion.

In this paper, ^we have studied the effect of the coupling ^{of a nematic on a Rouse} chain. This

can only ^{be a base for a more} realistic model quite suitable for the rheological behaviour of nematic polymers.

Thanks to our model, ^{we have} computed ^{the chain} geometry, ^{for each} mode, ^{in a} simple

shear flow, ^{for a} given ^order parameter. At low shear rate (and ^{in the case of} strong coupling

for instance), ^we have shown that the angle ^{of the} anisotropy ^{is no} longer equal ^to 77-/4, ^{on the contrary to the case of usual} suspensions. ^In simple ^{shear flow,} uncoiling ^of polymers ^could theorically ^{appear for very drastic} conditions. The first normal stress difference could be negative only ^{if the} angle between the nematic and the velocity ^is negative. This would require ^{a non linear} analysis, for instance by taking ^{into account the}

chain rigidity. The calculation here can be extended to get ^other physical quantities.

Finally, ^{the main} problem ^{is now to} define how the order parameter ^can be related to the coil geometry and the flow.

Acknowledgements.

We are grateful ^to J. F. Palierne and Y. Thiriet for helpful discussions.

References

[1] ^{ROUSE P.} E., ^{J. Chem.} Phys. ²¹ (1953) ^1272.

[2] ^{DE GENNES P. G., The} physic ^of liquid crystals (Pergamon press) ^1976.

[3] ^BROCHARD F., ^J. Polym. ^Sci. Polym. Phys. ^{Ed. 17} (1979) ^1367.

[4] D’ALLEST J. F., ^MAÏSSA P., ^et al., Phys. Rev. Lett. 61-22 (1988) ^2562.

[5] ^DOI M., ^EDWARDS S. F., ^The theory ^of polymers dynamic (Oxford University Press) ^1986.

[6] HOCQUART R., PALIERNE J. F., ^MULLER R., ^to be published.

[7] ^{For a} review, ^{see GOTSIS A.} D., ^{BAIRD D.} G., ^{Rheol. Acta 25} (1986) ^275.

[8] CHAFFEY C. E., ^{PORTER R.} S., ^J. Rheol. 29-3 (1985) ^281.

[9] ^MARUCCI G., MAFFETONE P. L., ^{Macromol. 22} (1989) ^4076.

[10] ^{For instance see in PIKIN S.} A., ^Sov. Phys. ^{J.E.T.P. 38} (1974) ^{1246 or in GUYON} E., PIERANSKI P.,

J. Phys. ^{France 36} (1975) ^203.