Rheology of Confined Polymer Melts under Shear Flow: Weak ...

+ PermanentMacromolecules 1996,28, 3901-3903 3901Rheology of Confined Polymer Melts under Shear Flow: WeakAdsorption LimitA. Subbotin,**t A. Semenov? G. Hadziioannou, and G. ten BrideLaboratory of Polymer Chemistry and Materials Science Center, University of Groningen,Ngenborgh 4, 9747 AG Groningen, The NetherlandsReceived December 8, 1994; Revised Manuscript Received March 13, 1995@1. IntroductionABSTRACT The dynamics of a confined polymer melt between weakly adsorbing surfaces is consideredtheoretically. The finite chain extensibility is taken into account explicitly, and the tangential stressand the first and the second normal-stress differences are calculated as functions of shear rate 9. Forsmall shear velocities (u -= u**) the surface slip is large, and the apparent viscosity, qapp, is proportionalto the layer thickness h and is independent of the shear rate. For very high shear rates, the surface slipis small and the tangential stress increases with velocity to the power V3. Alternatively, the apparentviscosity q decreases as a function of y with a characteristic -2/3 power law.Polymer-surface interaction becomes an importantparameter for polymer flow in a confined state whenthe characteristic size of the system is of the order ofthe polymer coil size. This fact is confirmed by anumber of experiments devoted to the dynamics of thinpolymer film~.l-~ In a number of cases, where theadsorption between polymer and surface is weak enough,surface slip isObviously, this surface slipdepends on the mobility of the polymer segments nearthe wall and is in general determined by the strengthof the polymer-surface interaction.In the present study we investigate theoretically therheological behavior of a thin polymer melt film confinedbetween two weakly adsorbing surfaces and subjectedto a shear due to a constant imposed velocity. Theopposite case of strong adsorbing surfaces has beencompleted re~ently.~ Most of the methodology is alreadypresented in the previous paper on nonlinear rheologyin the bulk state.1°As before, we assume the chains to consist of Nstatistical segments of length a and excluded volume u.The thickness of the confined film, h, is assumed to besmaller or equal to the size of the coils (h I UN'~). Thearea of contact per segment is ula, and an adsorbedchain has on average Nalh contacts with the surfaces.The polymer chains in a melt polymer layer locally obeyGaussian statistics.ll The global effect of the solid wallson the chain statistics can be accounted for by themirror-image principle.122. Nanorheology in the Confined State: WeakAdsorption LimitIn our previous paperlo bulk systems were considered,i.e. systems for which the distance h between the wallsgreatly exceeds the size of the polymer coil, h > aW".Now we consider the opposite case of a confmed polymermelt with h awl2. The polymer-surface interactionwill have a strong influence on the dynamics of thechains in this case. The mobility of polymer segmentson the surfaces and far away from it will be different.* To whom correspondence should be addressed.address: Institute of Petrochemical Synthesis,Russian Academy of Sciences, Moscow 117912, Russia.* Permanent address: Physics Department, Moscow StateUniversity, Moscow 117234, Russia.@ Abstract published in Advance ACS Abstracts, May 1, 1995.z4JYFigure 1. Characteristic conformation of the chain in theconfined state.In the middle part of the confined film the mobility ofthe chain segments is determined by a friction coefficient50, which in principle can be much larger thanthe bulk value 5. The motion of adsorbed chain segmentson the other hand is governed by a frictioncoefficient 51 different from 50, due to the surfacepotential with a typical scale -a. If the polymer-surface interaction is attractive, 51 is expected to begreater than the friction coefficient 5dCo =- 1. Here itis assumed that this condition is satisfied.It is convenient to consider a polymer chain as asequence of blobs of size h, each blob consisting of go -hzla2 segments. A finite fraction of the blobs (e.g. 4 2 )must form bridges between the surfaces (a blob is calleda "bridge" if it has contacts with both walls'l); eachbridge implies on the order of - hla contacts withthe surfaces. Other blods constitute loops (a loop has-hla contacts with one of the surfaces) (Figure 1).When a velocity u is imposed, the fraction forceimpacts both on the loops and on the bridges. In orderto calculate this force let us assume that the surfaceslip velocity is us, and therefore the shear rate insidethe layer is given byp = (u - 2UJh(1)We furthermore assume that the coil conformation canstill be described by the equilibrium statistics. Then,the characteristic friction force acting on a loop isfi - 5oYhgo (2)Obviously, this force, by means of the chain elasticity,causes a slip velocity us near the surface correspondingto the surface friction forcefl - 51ugo1'2 (3)0024-9297/95/2228-3901$09.00/0 0 1995 American Chemical Society

3902 Subbotin et al.Macromolecules, Vol. 28, No. 11, 1995“ t4 F 4 I 7R-aN11Figure 2. Characteristic conformation of a chain before andafter the transition point.From the condition f1 = fl the slip velocity is found tobeIn this regime, the slip is large, if the parameter ,B > 1 was investigated in ref 9. Forp

Macromolecules, Vol. 28, No. 11, 1995 Rheology of Confined Polymer Melts 3903InInXLFigure 4. Plot of tangential (uxz) and normal (NI) stressesand the apparent viscosity (rapp) as a function of shear rateYawconsidered as being independent from each other becausethere are no long-range correlations along thechain. Therefore, the tangential and normal stressesfor this case must be the same as in the bulk of chainscontaining go segments. Hence, the final results canbe obtained from the corresponding results in the bulk(ref 10, eqs 4 and 14) by substitutinggo for N and zo forzN, = 0The apparent viscosity, qapp, can be defined asEquations 17 and 18 imply thatwhereVapp = ax&/u (18)vaPp - p-2/3~01/3/~ u ’ u**u** -Note that u** >> u* and that in the region u > u** theslip is small (see eq 15b).The qualitative dependence of a, NI, vaPp as afunction of shear rate is shown in Figure 4. The first- u*, or 3 - l/(zag~~’~),normal-stress difference N1 has a jump in the point ubecause the chain changes itsconformation at this point from Gaussian to elongated.3. Discussion and ConclusionsIn this paper we present a theory for nonentanglementpolymer melts confined between two weaklyadsorbing surfaces. Our calculations show that thechain nonlinearity has profound influence on the surfaceslip behavior. The slip is large for small enough shearrates only, when the chains are still close to Gaussian.For high shear rates, the chains become stronglyelongated along the flow direction and, due to thenonlinearity, compressed in the normal direction. Thisfact leads to a decreasing surface slip.The principal difference between bulk and confinedmelt consists of the scale of the relaxation times.Although, a consistent description of relaxation processesin the confined melt does not exist yet, experimentsdo sho~l-~ that the relaxation processes arestrongly suppressed in the confined state. As a resultnonlinear behavior for confined polymer melts occursat much smaller shear rates, as manifested by theinverse proportionality of the critical shear velocity u*to the segmental relaxation time zo.Acknowledgment. This research was supported bythe Netherlands Foundation of Technology (SON STW)and the Netherlands Organization for Scientific Research(NWO). The authors are grateful to Dr. E.Manias for discussions.References and Notes(1) Gee, M. L.; McGuiggan, P. M.; Israelachvili, J. N.; Homola,A. M. J. Chem. Phys. 1990,93, 1895.(2) Homola, A. M.; Nguyen, H. V.; Hadziioannou, G. J. Chem.Phys. 1991,94, 2346.(3) Hu, H.; Carlson, G. A.; Granick, S. Phys. Rev. Lett. 1991,66,2758.(4) Granick, S. Science 1992,253, 1374.(5) Hu, H.; Granick, S. Science 1991,258, 1339.(6) De Gennes. C. R. Acad. Sci. Paris 1979.288B. 219.(7) Thompson,’P. A.; Grest, G. S.; Robbins,’M. Phys. Reu. Lett.1992, 68, 3448.(8) Manias. E.: Hadziioannou. G.: Ten Brinke. G. J. Chem. Phvs.I ,‘ 1994, 101,’1721.(9) Subbotin, A.; Semenov, A. N.; Manias, E.; Hadziioannou, G.;Ten Brinke, G. Macromolecules 1995,28, 1511.(10) Subbotin, A.; Semenov, A. N.; Manias, E.; Hadziioannou, G.;Ten Brinke, G. Macromolecules 1995,28, 3898.(11) De Gennes, P. G. Scaling Concepts in Polymer Physics, 2ndprinting; Cornel1 University Press: Ithaca, New York, 1985.(12) Bitsanis, I. A.; Ten Brinke, G. J. Chem. Phys. 1993,99,3100.MA9462060