Small angle neutron scattering studies of liquid systems in nonequilibrium

Physica A 174 (1991) 74-93 North-Holland

SMALL ANGLE NEtlTRON SCATTERING STUDIES OF LIQUID SYSTEMS IN NONEQUILIBRIUM

Peter LINDNER Institut Laue-Langevin, BP 156X, F-38042 Grenoble Cedex, France

Neutron scattering experiments in nonequilibrium will be illustrated with two typical examples from the field of polymer and colloid science. In particular, the coil deformation of singie, flexible polystyrene molecules in laminar shear flow and the shear induced demixing behaviour of polystyrene in a theta solution has been quantitatively investigated by this method. A turbulent flow experiment, on the other hand, revealed structural evidence of the orientation of rodlike miceUar additives under turbulent, drag reduced pipe flow conditions.

1. Introduction

1.1. Small angle scattering [1, 2]

In structural studies of polymers and colloids we are interested in the elastic coherent scattering process with approximately zero energy transfer (AE ~0). As a function of the momentum transfer during the scattering process Q,

47I" Q = I k - kol = -~- s i n ( ½ 0 ) , (1)

and the concentration P2 of the solute, the scattering intensity Y of a solution is observed:

d2 Y = A ~--~ (Q, p.,). (2)

k 0 and k are the momentum of the incident and scattered beam, respectively, A is the wavelength of the radiation and (9 is the scattering angle. A is a constant for a given experimental configuration and contains instrumental parameters (sample-to-detector distance L, primary beam intensity Y, and sample diaphragm area AA), as well as sample parameters (sample transmission T and sample thickness d). The differential scattering cross section dX/df2 per unit sample volume (d,~/d~ is usually given in absolute units of cm -1) is the probability of a quantum or particle of the primary beam with intensity Y0 to be scattered by a volume element of the illuminated sample into a solid angle

0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

P. Lindner / SANS studies of liquid systems 75

element dO of the detector. It is proportional to the scattering constant K, the molecular weight M and the single particle formfactor P(Q). P(Q) contains the complete information on the conformation (shape and size) of the dissolved particle [3]):

d,~ d--O (Q' P2)~ KMP(Q). (3)

The scattering constant K is proportional to the square of the contrast. The contrast itself is in the case of neutron scattering given by the differences of the scattering lengths b of the solute particles with respect to the solvent particles. The large difference in scattering length b for the hydrogen isotopes I H (b =-0 .374 x 10 -12 cm) and 2H (b = +0.668 x 10 -12 cm) is the reason why polymeric and colloidal systems with their large hydrogen content provide an excellent contrast in neutron scattering experiments when one of the components (either solvent or solute) is deuterated. It is assumed that isotopic substitution generally does not alter the chemical properties of the system.

1.2. Sample environment

In a standard small angle neutron scattering (SANS) experiment the (liquid) sample is usually confined in a quartz container of defined path length and is in thermodynamic equilibrium with its environment, at given chemical composi- tion (concentration), pressure and temperature. Without external constraints, an isotropic scattering behaviour is observed: the scattered intensity is under these equilibrium conditions a result of the space and time average of all molecular conformations and orientations.

In a nonequilibrium experiment [4], an external field is applied. External constraints can be imposed for instance by applying either a magnetic, an electric, or a hydrodynamic field to the sample. Furthermore, different modes of operation are possible: (i) thus, in a kinetic or relaxation experiment the sample is allowed to relax back to equilibrium after an externally imposed perturbation of its equilibrium configuration; (ii) in a cyclic experiment the sample is periodically distorted around its equilibrium state; (iii) a third type of non-equilibrium experiment is the steady-state experiment with a constant external input of energy.

1.3. Flow apparatuses for small angle neutron scattering (SANS) experiments

Neutron scattering experiments under nonequilibrium conditions can be performed at the small angle instrument D l l [6] of the Institut Laue- Langevin, Grenoble, France, using two types of apparatuses where the sample is exposed to shear flow with a transverse gradient 3~ = dv=/dx.

76 P. Lindner I SANS studies of liquid systems

1.3.1. Couette type shear apparatus [5] A transverse velocity gradient is realized between two parallel plates, one at

rest and the other plate moving with constant velocity v~ due to the action of a constant external force. In the case of the plane Couette flow the gradient ~, can be written as

d - const, (4)

where d is the distance between the two plates. A Couette type shear apparatus allows the SANS investigation of liquid systems in laminar shear flow (fig. la).

The sample container consists of an inner fixed piston (stator) and an outer rotating beaker (rotor), both made of quartz glass, which is highly transparent for thermal and cold neutrons and shows a very low small angle scattering background. The sample is confined in the annular gap between rotor and stator. The neutron beam hits the apparatus perpendicular to the axis of

beam

(a) =

perpendicular t

parallet

\ \

detector

beam adjustable (b) Cadmium diaphragm

pipe flow

Fig. i. Geometry. of the flow experiments. (a) Laminar shear [;.~w in the Coulette type shear apparatus with the .solution confined in the gap between both cylinders. (b) Pipe-flow geometry for the turbulent flow exr~er|ment (1: beam path centred to the pipe axis with centred diaphragm; 2: beam path only through near wall region with diaphragm in the lower position).


rotation and passes through both cylinders. The two cylinders have a wall thickness of 2.5 mm, their inner height is 60 mm and the inner diameter of the rotor is 48 mm. Depending on the stator, which exists in two versions (outer diameter 47 mm and 46 mm), the available cylindrical gap has a thickness of 0 .5mm or 1 .0mm, the sample volume being therefore V ~ 4 . 5 m l (gap= 0.5 mm) or V= 7.0 ml (gap = 1.0 mm). Within the stators there are two sepa- rate compartments at each side of the beam path, which are connected to an external thermostat for temperature control. A Pt-100 resistance element is fixed to the inside of the stator and records the temperature of the gap through the cylinder wall. For the given construction, the gap width is much smaller than the cylinder radius and to a good approximation a plane Couette flow is realized, with a constant transverse shear gradient ~,. The maximum attainable shear gradient depends on various experimental parameters and is a function of the viscosity of the sheared liquid [5]. For solution viscosities of the order of 10 mPas shear gradients of about ~, = 12 000s -1 can be obtained.

1.3.2. Turbulent pipe flow apparatus [7] When a fluid streams through a tube at constant flow rate (Poisseuille flow)

the velocity at the tube wall is zero and increases towards the centre of the tube. The parabolic velocity profile causes a non-constant transverse flow gradient, which linearly decreases from its maximum value at the tube wall to zero in the central tube axis. A turbulent pipe flow apparatus for SANS experiments has been constructed in collaboration with the University of Dortmund (fig. lb). It consists of a gear pump, a surge tank for damping fluctuations produced by the pump, a steel pipe of diaii~eter d - 15 mm and length 2.5 m and a heat exchanger for controlling the temperature. The pipe has an insertion made of quartz glass, serving as a window for the neutron scattering experiment. The neutron beam path is perpendicular to the pipe axis and passes through the quartz section transparent to neutrons. A slitlike cadmium diaphragm in front of the pipe is adjustable to different heights with respect to the pipe axis. The temperature is measured by a resistance ther- momenter, the pressure drop in the pipe by a pressure transducer and the flow rate by an inductive flowmeter. The total volume including the surge tank is about 3 liters. Maximum Reynolds numbers of Re ~ 50 000 can be obtained.

2. Polymer conformation in laminar shear flow

When polymers in solution are subjected to a hydrodynamic field, changes of the single chain conformation are to be expected. Investigation of the molecular dynamics and the structural changes on the microscopic scale are of course

78 P. Lindner / SANS studies of liquid systems

essential with respect to an understanding of the macroscopic material properties.

The changes of the single chain conformation of a dilute polymer solution in laminar Couette flow have been discussed already in 1943 by Kuhn and Kuhn for the case of an ideal flexible coil [8]. Due to viscous drag (friction forces) in the streaming solution, the whole molecule is undergoing combined rotation and deformation (fig. 2).

The elastic chain properties lead to a resistance against shape changes. Later, in 1945, Kuhn and Kuhn have treated "real" chains, where intramolecu- lar bond rotation should be hindered by local energy barriers. This (dynamic) effect should limit the deformation of the coil in shear flow [9]. Further theoretical approaches by several authors have dealt with the influence of a velocity gradient on the statistical chain conformation of polymers subjected to a hydrodynamic field, both for longitudinal [10] and transverse gradients [11]. However, experimental techniques like flow birefringence [12-14] are re- stricted in their information on the statistical conformation of the deformed coil because orly relative information on the shape changes is provided [16, 17]. More detailed information could be obtained if the polymer solution or melt is observed in a velocity gradient by scattering methods which give the overall coil size directly. Wide angle light scattering measurements of high molecular mass polyisobutylene solutions subjected to a Searle type shear field (inner rotating cylinder) have qualitatively confirmed the gradual deformation with increasing shear gradient [15]. Applying small angle neutron scattering to flowing polymers with the Couette type shear apparatus described above, most of the experimental difficulties and shortcomings of other methods can be avoided [5]. Quartz sample containers are highly transparent for neutrons and, in addition, this technique is able to resolve the Q-range of the interesting transitions in the scattering curve from the scattering of the overall coil to the scattering of the flexible segment and the monomer unit [3].

I

Fig. 2. Combincd deformation and orientation of a single chain molccule in laminar flow.

P. Lindner I SANS studies of liquid systems 79

2.1. Experimental

The linear flexible polymer studied with the SANS-shear experiments [16. 17] were fully deuterated, anionicaily polymerized polystyrenes with molecular masses Mw=160000g/mol, Mw=280000g/mol and M,,, = 500 000 g/moi. Considering the Q-dependent dynamics of a single polymer chain in dilute solution, experimental conditions could be estimated where a shear deformation of the coil can be expected. In view of the experimentally given maximum shear gradient of Tmax = 12 000 S-~ it turns out that the intersegmental motion of the polystyrene chain has to be slowed down by choice of an appropriate viscous solvent, in order to shift the space-time behaviour of the polymer coil into the time domain of SANS in shear flow [16]. The mixture of commercially available oligostyrene-h, (OS, Pressure Chemical, M,,,=800g/mol, M,,,/M. = 1.3) with toluene-h 8 is a suitable viscous solvent for polystyrene. The

combination of the deuterated polymer with a protonated solvent provides a sufficient scattering contrast for neutron scattering experiments.

For each molecular mass of the polymer a concentration series has been investigated at rest (~, = 0 s -I) and in constant shear flow with several shear- rates up to q = 8 5 0 0 s -~ in a range of momentum transfer 0.05< Q < 0.20 nm-1. The small angle neutron scattering intensity of the sheared polymer solutions shows a more or less pronounced anisotropic intensity distribution on the 2D-multidetector with elliptical isointensity lines (fig. 3a,b). The magnitude of the effect depends upon molecular mass, shear gradient and solvent viscosity. The raw data of the isotropic spectra (polymer solutions at rest, pure solvent) are radially averaged and subsequently corrected for sample container- and background-scattering. The corrected scattering intensity is normalized with the scattering of a 1 mm water sample as standard reference. Subtraction of the normalized solvent intensity yields the absolute intensity

i - • A ~ , b ) ~ ".,. ~:

• z o . : - ,

' ; " ' I

0 n o

Fig. 3. S A N S raw data of a polystyrene solution of concentration P2 = 8 m g / c m 3, solvent viscosity

?7 = 50 mPas. (a) at rest; (b) in a shear gradient ~, = 6000s -1.

80 P. Lindner i SANS studies of liquid systems

(differential scattering cross section of the polymer) d 2 / d ~ as a function of scattering vector Q and polymer concentration P2 [30]. The anisotropic SANS intensity of the sheared solutions is evaluated in the directions perpendicular ( 1 ) and parallel (!1) with regard to the flow direction in the gap of the shear apparatus. The conformation parameters of the single polymer chain, i.e. the radius of gyration ( R ) , (as a measure of the overall coil size), the molecular mass M,,. and the second osmotic virial coefficient A 2, are obtained from square root plots of the differential scattering cross section d ,~/d~ at low Q as

KNP2 )05 (d .~ /dn) = f(const x Q2 + p2) (5)

(with K~ the characteristic contrast parameter for SANS [3]) and extrapolation to concentration P: = 0 and momentum transfer Q = 0.

2.2. Results

Fig. 4a-c shows as an example three square-root plots of the concentration- and Q-dependent differential scattering cross sections of four solutions of polystyrene with M,,. = 280 000 g! mol. Fig. 4a results from the evaluation of the isotropic scattering intensity of the solution at rest (j, = 0 s- i ) . Fig. 4b, c shows the differential scattering cross sections (anisotropic scattering patterns) of the sheared solutions in a shear gradient j, = 1500 s -l, evaluated in the directions perpendicular (± ) and parallel (11) with regard to the flow direction.

The following results are obtained for the conformation parameters: (i) as expected, the molecular masses M,,. measured with SANS are equal to those known from the GPC measurements; (ii) the second osmotic virial coefficients A 2 resulting from the slope of the concentration-dependent line after extrapolation to Q = 0 are positive, i.e. the solvent mixture oligostyrene + toluene is a thermodynamically "good" solvent for polystyrene; (iii) in contrast, a signifi- cant anisotropy of the experimental radii of gyration (following from the slope of the differential scattering cross-section d ,~ /d~ after extrapolation to concentration P2 = 0 ) with regard to the directions i and II on the detector is observed in the case of the sheared solutions. Fig. 5 shows the eperimental radii o f gyrat ion ( , , , , , , /~, , , , ,u, , , , ,~, , ~ j , \ " I t / ~ ' " ~ " ~ ' ~ .~,,,uu,,,,. ~v,,,u,,uo,, in

direction parallel with regard to the flow) and (R~) (sheared solution, evaluation perpendicular) as a function of the shear gradient ~, for polystyrene M,~. = 280000g/mol in the solvent oligostyrene + toluene (Wos =0.735, , /= 50mPas). The value (RII) clearly increases with increasing shear gradient ~, whereas the value (R~) within experimental error is identical to (Ri~,,) and independent of j,. The anisotropy of the scattering pattern of the flowing


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15

Fig. 4. Square-root plots of the concent ra t ion- and Q - d e p e n d e n t differential scat ter ing cross sect ions of po lys ty rene (M,~ = 280000 g / m o l ) in absolute units ( c m - t ) . (a) Solution at rest; (b) shea red solut ion, da ta evaluat ion in the d i rec t ion perpendicu la r to the flow (5' = 1 5 l l 0 s - ' ) : (c)

shea red solut ion, para l le l to the flow (5' = 1500 s - ' ) .

82 P. Lbzdner / SANS snldies of liquM systems

32

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1 0 0 0 0 1 2 0 0 0 0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0

shear gradient [ s ' t l

Fig. 5. E x p e r i m e n t a l radi i o f g y r a t i o n f r o m the so lu t i on a t res t ((R,,,,}) and f r o m the s h e a r e d so lu t ions ( ( R ) . ( R I , ) ) as a f u n c t i o n o f shea r g rad ien t ~,.

solutions indicates already qualitatively the shear-induced deformation of the polymer chain.

2.3. Comparison of experimental results with theoretical predictions

A deformation ratio can be be defined as the ratio of the averaged squares of the radius of gyration in flowing solution to the (isotropic) radius of gyration of the molecule in solution at rest: deformation ratio = (R 2 } / (R~,,,} - 1. Fig. 6 shows the deformation ratios resulting from the SANS-shear experiment [17] in conjunction with other experimental results [15] in a double logarithmic plot of (R- ' ) / (R~, , , ) - 1 as a function of the reduced gradient/3. /3 is a generalized, dimensionless orientation variable and serves as a suitable parameter for

,° I ~ / / /

I / / / / /

/ / • Q®

e~g '

0 . 1 / / "~ [ / / "

i t / / / / /

0 . 0 1 ( z . . . . . . . i . . . . - - - ~, 0 . 1 I I 0

P

. . . . . . | . . . . • . . . . I . . . . . . . .

R o u s e model / Z i m m model i /

o~ 0.237 • t 32 / /-4------ 0 : 0 . 1 3 6 • 132 / /

1 , • . o . , , - ~ -

1 0 0

Fig. 6. E x p e r i m e n t a l d e t o r m a t i o n r a t i o (R:}/(R~,,} - I as f u n c l i o n o[ r educed s h e a r g r ad i cn t 13, t oge the r ~'itth m ~ e l ca lcu la t ions [18].

P. Lindner / S A N S studies o f liquid systems 83

different experimental conditions:/3 = ([~I]71Mw~/)/RT , with [r/] = intrinsic viscosity, r /= solvent viscosity, M w = molecuar mass, ~ = shear gradient, R = gas constant, T = absolute temperature. Also shown in fig. 6 are the results of approximative calculations of Peterlin for an ideal flexible coil with ("Zimm behaviour") and without ("Rouse behaviour") consideration of the hydrodynamic interaction [18]. A common feature of the theoretical models is the variation of the mean chain-extension with /32, the only difference between both models being the prefactor C ((hE)/(h,2~) -- 1 = C/3 2 where (hE), (h 2) is the mean square end-to-end distance of the coil under shear flow and at rest, with C - 0 . 2 6 7 for the Rouse model and C =0.136 for the Zimm model).

According to the experimental results, the deformation ratio of the polystyrene chain follows the f/E-dependence characteristic for the ideal flexible molecule only at low/3-values (/3 < 1). At larger/3 the mean chain-extension increases less than predicted for a perfect flexible coil. Employing theoretical calculations [19], this result can be interpreted in terms of the internal viscosity concept: generally, the deviations at larger /3 are to be expected since the above-mentioned calculations for the chain-extension of the ideal flexible coil with unhindered bond-rotation under shear flow do not consider the finite "dynamic" rigidity of the molecule (complementary to the "static" rigidity of the chain as defined by the Kuhn statistical element). The limited dynamic flexibility of the real polymer chain due to hindrance of free rotation of the chain segments by activation energy barriers leads with increasing gradient/3 to an increasing resistance against deformation. Hence, the deformation ratios at large /3 should be lower than those for a dynamically infinitely ~iexible molecule, which is in agreement with the experimental findings [17].

2.4. Results at high Q-values

The influence of the hydrodynamic field on the length scale of the statistical chain element was also investigated at higher Q-values [20]. Fig. 7 shows the polystyrene scattering curve up to Q = 1.8 nm -~, measured in a shear gradient of ~, = 1500 s-1 and evaluated in the direction parallel and perpendicular with regard to the flow direction. The anisotropy of the differential scattering cross section is decreasing with increasing momentum transfer. The short range distribution ef the chain elements (high-Q region) is less perturbed by the shear gradient than the overall coil size on length scales of the order of the radius of gyration (Iow-Q region).

2.5. Sheared dilute polymer solutions in a theta solvent

Recently, SANS experiments under sheared conditions have been performed with a polymer solution at the theta point [20]. A commercially available

84 P. Lindner / SANS studies of liquM systems

j o ~

lOs

squares: .1_

t r i a n g l e s : ] [

1¢

O'

1 0 0 @ ' " " - " ' ' " . . . . . . . . ' . . . . . . . . 0 . 0 0 1 0 . 0 1 O . 1 1

Q IA'tl

Fig. 7. Differential scattering cross section of polystyrene Mw = 280 000 g/mol in a shear gradient of ~, = 6000 s-t. The draw'- line corresponds to the scattering curve measured at rest.

polystyrene of molecular mass M,,,= 230000g/mol (Pressure Chemical) was dissolved in fully deuterated dioctylphthalate (DOP-d, viscosity ~7 ~" 74 mPas) as solvent. The theta point of the system (with protonated components) is reported in the literature as T = 22°C [21]. In the case of the present system (deuterated solvent) a theta temperature of around T= 34°C is found. The measurements were done at the instrument D l l in a Q-range of 0.05 ~< Q ~< 0.45 nm -j, using the Couette type shear apparatus at rest and at a shear gradient of ~/= 104 s-

in contrast to the shear experiments performed on polystyrene in the thermodynamically "'good" solvent oligostyrene + toluene (positive second osmotic virial coei'ficient), where an increasing coil deformation has been found with increasing shear gradient, a complete different feature appears in the case of the sheared theta solution. In comparison to the solution investigated at rest, the sheared solution also shows an anisotropic intensity distribution on the two-dimensional muitidetector at a gradient of ~, = 104s -~. Treatment of the data in the two directions perpendicular and parallel with respect to the flow direction revealed, however, an increase of the scattering intensity in both directions ± and ii at low Q-values (fig. 8).

Furtherm.:~rc, when removing the sample from ~he Couette cell after the shear experiment, the solution was optically turbid. At present, this result is interpreted as a consequence of a solvent quality change under shear. Hence, due to the actton of the hydrodynamic field, the theta temperature of the system is virtually increased. The turbid appearance of the sample directly after the shear ex~r iment suggests shear induced demixing. The observed anisot-


106

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squares: .1_ & i n

a triangles: II I ftAoo.

: rest

! | ! t ! t t t I ! ! i t ! t " "

0 .001 0.01 0.1 Q l~ ' t i

Fig. 8. Differential scatte, jg cross section of polystyrene M,, = 230000g/mole, measured in a theta solvent (di-octylphthalate) at rest and in laminar shear flow in a shear gradient of ~, = 104 s '.

ropy of the scattering pattern of the sheared solution at low Q indicates that anisotropic aggregates are formed under shear.

The previously sheared solution was kept at ambient temperature and has been subsequently investigated under static conditions at D l l in a Q-range 0.014~< Q <~0.45 nm -~ and at different times during a long period of several months. Fig. 9 shows the results as differential scattering cross section as a function of momentum transfer, together with the measured scattering curve of the single polystyrene chain in solution at rest before the shear experiment a,ad

V , I

,°° f 10

0 I l " 1 0 " 4

. . . . . . . . I . . . . . . . . I . . . . . . . .

0 o~----- 1 month later

o o 1 week later

0 o 4[""

% % o o ~ immediately after 7 months!ater ~ experiment

before_ " ~ shear experiment

\ • • • • . . a . . , I . . . . . . . . I . , , . . - ' ~ -

10 ":~t tu I 10-1

Q [~z tl

Fig. 9. Differential scattering cross section of polystyrene M,, = 230 000g/mol measured at rest and at different times after cessation of shear. Drawn line: Debye fit for the polystyrene scattering curve measured at rest, before the shear experiment.

86 P. Lindner ! S A N S studies of liquid systems

the corresponding fit of the Debye function. The strong increase of the scattering intensity at low Q-values disappears on a time scale of about 7 months. On the length scale of the radius of gyration of the single polymer chain a gradual dissolution back to the original conformation is observed.

3. Drag reducing surfactant solutions in laminar and turbulent flow

Drag reduction is the reduction of the friction factor in a turbulent flow at a constant flow rate. It can be achieved by addition of high molecular weight polymers, surfactants or fibres to a Newtonian fluid like ware," at lowest concentrations (Toms" effect [22]). Result," p,a01ished so far suggest that drag reduction with surfactant solutions can or~'y occur when the solution contains rodlike micelles [23]. However, a satisfactory description of the interaction between turbulent eddies and the additive which leads to the drag reduction is still missing. Intuitively, one would perhaps expect a transition from random orientation of the anisotropic particles at rest towards an alignment in laminar flow and, with increasing flow velocity, again a random orientation in the turbulent flow regime (fig. 10).

Suffactant solutions containing rodlike micelles reveal a characteristic friction behaviour when the pressure drop of such a solution (being proportional to the friction factor f ) is measured under laminar and under turbulent pipe flow conditions as a function of the flow velocity (being proportional to the Reynolds number Re). The friction factor f of the surfactant solution is defined as f = Ap dl2pw21, where Ap means the pressure drop over the pipe length l, d the pipe diameter, p the density of the fluid, and w its bulk velocity. The Reynolds number Re, which is based on the solvent viscosity r/, is defined as Re = wdp/rl. The friction factor f as a function of the Reynolds number Re

at resl

laminar flow (above critical shear gradient) turbulent

J

isotropic scattering random o, ientation

anisotropic; isotropic; alignment random

orientation

Fig. i0. Expectation of orientation of rodlike micellar particles in different flow situations


(fig. 11) has been measured with the turbulent flow apparatus described above (fig. lb). Three characteristic regions can be distinguished in fig. 11:

(i) In the laminar flow region (A) the friction behaviour of a Newtonian fluid is described by the Hagen-Poisseuille law f = 16/Re.

(ii) In the intermediate turbulent regime the friction factor f of the solution is substantially lowered with respcct to the friction factor of the pure solvent (drag reduction). Virk [24] has shown by an analysis of available literature data that the limit of the maximum attainable drag reduction is given by 1/X/~ = 19 log(Re V ~ ) - 32.4 (lower drawn line in fig. 11).

(iii) Exceeding a critical wall shear stress at higher Reynolds numbers Re, the drag reduction starts to break down and the friction factor of the solution approaches the friction behaviour of a turbulent Newtonian liquid. The turbulent flow region of a Newtonian fluid is described by the Prandtl-Karman law [25] 1/V~ = 4 log(Re V ~ ) - 0.4 (upper drawn line in fig. 11).

Shear viscosity measurements of the surfactant solution in laminar Couette flow reveal a sudden increase of the viscosity above a critical shear rate ~,*. The effect is immediately reversible. Above the threshold shear rate ~/* the solution shows viscoelastic behaviour. This feature has been found with different surfactant solutions containing rodlike micelles and has been attri- buted to a so-called "shear induced state" (SIS [24, 26-28]). A simple shear alignment of long rodlike particles should lead to a decrease of the viscosity with increasing shear rate. Therefore shear induced new structures have to build up in the SIS.

t t .

t _

o

, m

°-1 I 0.01

o.oo11!

• " ~ . . . . I

A B

i |

C

i , , * i i I

10 4 I , i i i i " " *

11

Reynolds number Re

Fig. 11. Friction factor f of a surfactant solution (cf. section 3.1) as a function of Reynolds number Re. Points: fresh solution; triangles: previously stressed solution; A: laminar flow region" B: turbulent, drag reduced flow; C: turbulent flov: region.

88 P. Lindner I S A N S studies of liquid systems

In order to study the molecular basis of drag reduction under real flow conditions, the conformation of rodlike micelles has been investigated by small angle neutron scattering experiments [35] with a surfactant solution subjected to laminar Couette flow and turbulent pipe flow (exhibiting drag reduction).

3.1. Experimental

The investigated drag reducing additive was an equimo!ar mixture of the cationic surfactant tetradecyl-trimethyl-ammoniumbromide (C ~ 4 H 29- N(CHa)3)Br with sodiumsalicylate (C6H4OHCOONa) dissolved in heavy water (D20). The total concentration of the solute was 2.4 mmol/l corresponding to about I g /kg (1000ppm w/w). At a temperature of 20°C and above the critical micellar concentration Ccm¢ = 0.25 g/kg [29] this system assembles into aggregates of very long rodlike micelles (called C~aTAB/NaSal-h ) with a large aspect ratio (ratio length to diameter). In order to allow for an equilibrium conformation of the micellar system the solution was heated up to T ~ 60°C for 12 hours immediately after the preparation and was then cooled down to the measuring temperature of T = 20°C two days before the experiment (fresh solution).

3.2. SANS measurements in laminar and turbulent flow

The SANS experiments were performed at the instrument D I1 [6] of the Institut Laue-Langevin in Grenoble, France, with an effective range of the momentum transfer of 0.08<~Q<~0.75nm -~. Background correction of the solution spectra was done with the sovent- and the empty-cell measurement and the data were calibrated with a standard water sample of thickness i mm to allow for calculation of the differential scattering cross section of the solute (d,~/d.f/) in absolute units (cm -~) [30]. During the SANS experiments the temperature was kept at (20.0-+-0.1)°C.

The surfactant solution was measured under laminar shear flow conditions in the Couette type shear apparatus described above (fig. l a) using concentric quartz cylinders with a gap width of d = 0.5 mm at different shear gradients in the range 75<~ 5'<~4000s -~. The SANS experiment under turbulent, drag reduced flow conditions at Reynolds numbers in the range 850 ~ Re <~ 23220 was pe,~uim~u w,t. the turumen[ flow apparatus (see above) which r, as aiso been used for the measurement of the friction behaviour. The neutrc, n beam path was perpendicular to the pipe axis and passes through the quartz section transparent to neutrons (fig. lb). The slitlike cadmium diaphragm in front of the pipe is adjustable to different heights with respect to the pipe axis. In particular two positions of the di.",phragm were used: (i) the centred position with the beam passing through the centre of the pipe and (ii) a lower position


with the beam passing mainly through the near wall region of the pipe. In addition, SANS measurements of the fresh solution, the stressed solution and the solvent D20 were done at rest using standard HELLMA quartz cells with a thickness of d = 5 mm at temperature T = 20°C, covering an effective Q-range of 0.028 ~< Q <~ 1.5 nm-~.

3.3. Results

Fig. 12 shows the results of SANS measurements of the surfactant solution in laminar shear flow, at various shear gradients ~,. The results are presented as isointensity-contour plots of the 2D-multidetector data. With increasing shear gradient ~, the anisotropy of the scattering pattern increases. This corresponds to an increasing alignment of the rodlike particles with their long axis parallel to the direction of flow. However, as a comparison with calculation [31] shows [29] the obtained alignment cannot be explained by a rod length of L = 248 mm (cf. ref. [32]) and suggests an increase of the average rod length by a factor >10.

Results of the SANS measurements using the turbulent flow apparatus at various Reynolds numbers Re under turbulent, drag reduced flow conditions show a similar increasing anisotropy with increasing Reynolds number as in the Couette experiment under laminar shear flow. This proves that even in turbulent, drag reduced pipe flow the rodlike micellar particles are increasingly oriented parallel to the flow direction. The experimental findings are under those conditions (turbulence) in contradiction to the expectation of a random orientation (fig. 10c).

Fig. 13 shows the contour plots of the surfactant soludon in turbulent flow at the three highest measured Reynolds numbers (Re = 12380, 18890 and 23220) for the two experimental geometries where (i) the neutron beam mainly monitors the core region (left column) and (ii) the near wall region of the pipe (right column).

However, a more pronounced anisotropy is observed when the neutron beam passes mainly through the near wall region of the pipe. Moreover, in the breakdown region of the drag reduction (Re>~ 15000) the anisotropy of the scattering pattern clearly decreases with increasing Reynolds numbers for both

t l - 41- ° beam paths (core and near wall region). These . . . . '+ ,,,,.r,,,t,~,t le~u,~s prove ,ha, In ~.. . . . . . . . . , drag reduced flow a difference in ordering of the rodlike particles exists between the near wall region and the core region of the pipe in agreement with the differences of the turbulent velocity profile between near wall and core region found with Laser Doppler Anemometry (LDA) measurements [34]. Furthermore, the breakdown of drag reduction accompanied with a breakdown of the orientation of the rodlike particles is confirmed [33].

90 P. Lindner I SANS studies of liquid systems

- ~ r - ; , l . • • . " + ~ l 4H~e I • ._t:, o o_11 o ~ " ~ " • 0

• , , o o o < ) I

,+ ,i,. ,,,o + .

" + + o~.+~. N I I ~ _~<9 .+m . ° ~ . k - ' ~ t - . . . , . J . / ' f . . . ~ ° I

• c : l

- - * A ~ J

"'6+('g I I ,~q~. '" l - ' ),.2.~ . b - - - - + d, f o . l

o " . , ~b ,

° , ~ o . " •

Q

0 O

,o o •

> 0 "

"~/s-' = 1 0 0 0 . ' . 0 " ~ . . . . . . . . . -

"+ • "~ "s,"-'O"t¢

Fig. 12. isointensity contour plots of the normalized S A N S multidctector data ( D I I , L = 5 m, a = 0 . 6 n m ) for the surfactant solution (2.4retool/! C,+TAB/NaSal -h in D , O at T = 2 0 ° C ) in laminar +~hear flo~;" at various shear gradients 3, (s ').


beam path centred to pipe axis

D

• .

0

R e = 1 2 3 8 0 ~ %

0

O• C~

R e = 1 8 8 9 0 ~

o

t

e=232 o o

beam path through near wall region "~° . -&

t

• ~ 0

Re=12380 0

$

4 '

, O

• ~o ~ ~ - ~ "

o

• o0.

!

o i I ,o=,~,,0~'.~ IIl

Fig. 13. lsointensity-contour plots of the normalized SANS multidetector data ( D l l , L = 10m, A =0.6rim) for the surfactant solution (2.4mmol/l C,4TAB/NaSal-h in D20 at T= 20°C) in turbulent pipe flow at the three highest measured Reynolds numbers Re. Left column: beam path centred with respect to pipe axis; right column: beam path through the near wall region.

92 P Lindner ! SANS studies of liquid systems

4. Conclusion

In a nonequilibrium scattering experiment the sample is subjected to an external field. A particularly interesting combination is the steady-state flow experiment where the (liquid) sample is exposed to a hydrodynamic field [5]. D~,dng the scattering experiment it experiences forces which tend to orient and to deform it. With respect to the two-dimensional multidetector the intensity distribution becomes usually anisotropic and important information is for instance obtained about the shal~ ~ and mobility of flexible or anisotropic particles. Neutron scatering is especially suited for these investigations due to the short wavelength of neutrons and hence the possibility to collect the scattered radiation close to the primary beam at small angles, which facilitates the construction of flow devices.

It is obvious that an "on-line" scattering study of flowing systems gives insight into a very realistic state of matter since many natural phenomena (e.g. blood circulation) or industrial processes (e.g. polymer fibre- and foil-extru- sion) involve such nonequilibrium rather than equilibrium conditions. Although the macroscopic features are well known from rheological measurements (e.g. viscoelasticity), the underlying mechanisms on the molecular scale have until now not been completely understood. Thus, a unique method for relating changes of the microscopic structure to macroscopic bulk behaviour is provided.

References

ill 121 131

141 151 161 171

181 i•1 I ' 1

Ii01 fell 1121 1131 1141 1151 1161 !171

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[351

A. Peterlin, J. Chem. Phys. 39 (1963) 224. R. Cerf, J. Chim. Phys. 66 (1969) 479. P. Lindner and R. Oberthiir, unpublished results. P. Stepanek and C. Konak, Coll. Czech. Chem. Comm. 50 (1985) 257 ~ B.A. Toms, Proc. 1st Int. Congr. on Rheology, vol. 2 (North-Holland, Amsterdam, 1948) p. 135. D. Ohlendorf, W. Interthal and H. Hoffmann, Rheol. Acta 25 (1986) 468. P.S. Virk, AIChE J. 21 (1975) 625. L. Prandtl, Z. Ver. Dtsch. Ing. 77 (1933) 105. H.W. Bewersdorff and D. Ohlendorf, Colloid Polym. Sci. 266 (1988) 941. M. L6bl, H. Thurn and H. Hoffmann, Ber. Bunsenges. Phys. Chem. 88 (1984) 1102. H. Rehage and H. Hoffman, Rheol. Acta 21 (1982) 561. P. Lindner and R. Oberthiir, unpublished results. M. Ragnetti and R.C. Oberthiir, Colloid Polym. Sci 264 (1986) 32. J.B. Hayter and J. Penfold, J. Phys. Chem. 88 (1984) 4589. J. Kalus, H. Hoffmann and K. Ibel, Colloid Polym. Sci. 267 (1989) 818. H.W. Bewersdorff, B. Frings, P. Lindner and R. Oberthiir, Rheoi. Acta 25 (1986) 642. H.W. Bewersdorff, in: Structure of Turbulence and Drag Reduction, A. Gyr, ed. (Springer, Berlin, 1990). P. Lindner, H.W. Bewersdorff, R. Heen, P. Sittart, H. Thiel, J. Langowski and R. Oberthiir, Progr. Colloid Polym. Sci. 81 (1990) 107.

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Small angle neutron scattering studies of liquid systems in nonequilibrium