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Viscosity of sponge phase in porous medium
C. Vinches, C. Coulon, D. Roux
To cite this version:
C. Vinches, C. Coulon, D. Roux. Viscosity of sponge phase in porous medium. Journal de Physique II, EDP Sciences, 1992, 2 (3), pp.453-469. �10.1051/jp2:1992143�. �jpa-00247643�
Classification Physics Abstracts
47.55M 51.20 82.70D
Viscosity of sponge phase in porous medium
C. Vinches, C. Coulon and D. Roux
Centre de Recherche P. Pascal, Av. Dr. Schweitzer, 33600 Pessac, France
(Received 25 July 1991, revised 18 November 1991, accepted 21 November 1991)
Abstract. Sponge phases are phases of randomly connected membranes. We present an
experimental investigation of their rheological properties. Both data on the « bulk » viscosity and the effective viscosity through pores are presented. When the bulk viscosity is of the order of a few times the solvent viscosity and is Newtonian in the accessible shear rates (10~~-1Iis~ ~), the flow of the sponge through cylindrical pores leads to non-Newtonian behaviour. At low shear an enhanced viscosity as the sponge size approaches the pore diameter is observed, while at larger shear rates, the apparent viscosity saturates towards a smaller value close to that measured in the bulk. A simple model based on obstruction and deformation is introduced to describe these
different regimes.
Introduction.
Studies of viscosity of fluids in porous media have both fundamental interest and many
important practical applications as for example in oil recovery [I]. In the simple case of flow
through cylindrical pores, unusual rheological properties are expected when the characteristic
length of the fluid becomes comparable to the pore diameter. A well known realization of this situation has been described for solutions of polymers where large characteristic lengths are
accessible either in the dilute (radius of gyration) or semi-dilute regimes (mesh size). In this limit different properties are observed depending on the shear rate [2-5]. At low shear, when the characteristic size of the polymer becomes of the order of the pore size, the flow is
impeded. This behaviour is called
« obstruction » effect in the following. Deformation of the
polymer is eventually expected at higher shear leading to a non-Newtonian behaviour. When the polymer size becomes larger than the pore diameter, a pressure threshold is predicted
below which no flow occurs [2, 3].
Surfactant molecules in solution are known to form aggregates of various shape and size.
Recelitly, a great deal of experimental and theoretical interest has been focussed on two- dimensional aggregates (membranes). These bilayers can either pack in long range ordered lamellar phases [6] or lead to isotropic liquid phases of randomly distributed membranes : the so-called sponge phase [7]. In this latter phase, the membrane separates the space into two
equivalent regions of solvent [8]. From simple geometrical arguments it can be shown that the characteristic size f of the sponge, which is the typical distance between membranes, increases with dilution as I/~b~ (~b~ being the membrane volume fraction) [7]. Thus, large dilution
JOIJRNAL DEPHYSIQUE II -T 2, N'3, MARCH 1992 IS
allows the preparation of samples with large f (100-1 0001) and the sponge phase is a good candidate in which to observe the obstruction effect. We describe in this paper such an observation where microporous filters are used to model the porous medium.
The first part of the paper is devoted to sample description. X-ray scattering results are
given in a second part. The third part deals with bulk viscosity measurements. Flow
experiments through pores are described in the last part and a simple model is proposed to analyze the data.
Sample preparation.
The phases of interest in this study are obtained with quatemary mixture of sodium dodecyl
sulfate (SDS), water, pentanol and dodecane [9]. Figure I shows the relevant phase diagram
in which a sponge phase, a lamellar phase and a microemulsion domain are present. The
sponge phase is constituted of inverted membranes (water film surrounded by a surfac-
tantlalcohol layer) and is found for a large domain of oil concentration (Si and in an isolated island (S~). Both domains allow the preparation of dilute samples. In the first domain, the sponge can be swollen without changing the constitution or the composition of the membrane
following a «dilution line» as shown in figure I. We have prepared samples with
compositions ranging between 50 and 90 9b oil. Higher dilution is prevented by a first order
phase transition with a phase of practically pure solvent (oil and alcohol). However, more dilute samples can be prepared in the island (S~). Each sample can be characterised by its
membrane volume fraction. Because W/S, the water over surfactant weight ratio, is kept fixed all along the dilution, we can characterize the sample with the water volume fraction ~b which is proportional to the membrane volume fraction. Samples with ~b ranging from 20 to 1.5 9b
were prepared in Si. Moreover a very dilute sample was prepared in S~, it corresponds to a
sponge with ~b
= 0.53 9b.
pentanol
I
L~
s W/S
= 1.20 dodecane
~
S
Fig. 1. Phase diagram of the quatemary system from which the sponge phases are prepared. The domains Si and 52 mentioned in the text are shown. The dashed line shows the
« dilution line » followed in St.
X-ray scattering measurements.
In order to characterise the samples, X-Ray scattering measurements have been performed.
Previous results being available only in the plane W/S
= 1.5 [7], we have extended these
measurements in the present case (W/S
= 1.2). Figure 2a shows typical results for the X-ray
1 (u.a)
I
I '
a b
i
I
,
o.oo
q 1 ~ o-i
q
ig. 2.
dilution line = 18.77 fb dangles and 14. I fb squares) ; (b) reduced intensity (after ormalisation
the water olume against wave vector in a plot for amples along the dilution
line ~ j = 18.7 fb, #~ =
9.7 fb, #~ = 3.9 fb). Straight lines with
shown.
scattering intensity as a function of the wave vector q =
(4aT sin 0/2)/A, where 0 is the
scattering angle. A maximum of I (q) is observed at a wave vector qo. As expected from the structure of the sponge, this correlation peak shifts towards the low q region as the membrane
concentration decreases. Figure 2b gives the reduced absolute scattering In (I/~b ) for three
samples along the dilution line for comparison of the different regimes [7] :
For q vectors larger than the peak position (q ~ qo), all curves have a universal behaviour characteristic of the form factor of lamella. This domain can be divided in two parts. A
I/q~ behaviour is found when q < 2 ar/8 (8 is the bilayer thickness) which is expected for the
scattering of a flat bilayer with random orientation, while a I/q~ behaviour characteristic of the scattering from a thin interface is reached when q ~ 2 ar/8. This latter behaviour is known
as the Porod limit.
For small wave vectors, I-e- when q <qo, departure from the universal form factor is observed due to correlations between lamella and is therefore sensitive to the membrane
volume fraction. In fact light scattering experiments have shown that concentration
fluctuations are present at small q in the sponge phase with a specific correlation function [8, 10].
The characteristic length f of the sponge can be obtained from the peak position
qo as qo =
2 «If [7]. This characteristic length is the membrane/membrane typical distance and corresponds to what we call the sponge size. Figure 3 shows the variation of the
characteristic length f obtained from the peak position as a function of the inverse of water volume fraction. A linear behaviour is found while the characteristic length varies over the range 100 to 3001. This is in agreement with the simple relation f
= a 8/~b (a is a numerical factor) which is usually obtained when logarithmic corrections are neglected [7, 11, 12].
Above 3001extrapolation is used to obtain the characteristic size (300-1600 A).
The most dilute sample (prepared in S~) has a characteristic length too large to be measured
by X-ray scattering but it can be measured using light scattering. In this case the correlation
peak can be located and leads to f m 6 0001 [13]. Note that this length is in agreement with the extrapolated value.
11
s lo
Fig. 3. Sponge size deduced from the X-ray correlation peak as a function the inverse of the water volume fraction. The straight line corresponds to an effective thickness «8
=
241.
In summary, these results are similar to previous studies and support the structural
description of the sponge phase as an isotropic phase of multiconnected membranes with a characteristic length f proportional to I/~b.
Bulk viscosity measurements.
We have measured the bulk viscosity of samples along the dilution path shown in figure I.
Measurements have been done upon steady shear in a viscometer with a Couette geometry.
Two different types of viscometer have been used, I-e- a Controvers (Low 30) and a
Rheometrics (RFS 8400) to explore a large range of shear rates (10-2 to 102 s-I). Figure 4a shows the variation of the stress « as a function of the shear rate f for three typical samples taken along the dilution line in St. For all samples the ratio «If gives a bulk viscosity independent of the shear rates, I-e- a Newtonian behaviour as shown in figure 4b. In the same
(1)
~
° (2) mPa
~~~
(3)
~
~fi
mPa.s
a b
__
0 10 20 30 40 50 60
~j j j ~~ j~j~
o o
'f s 'f
s .1
Fig. 4. (a) Measured stress as a function of the shear rate for three samples of the dilution line (the
water volume fraction are 11 = 10.5 fb, ~~ = 3.06 fb, ~~
=
2.28 fb). (b) Log-Log representation of the
stress as a function of the shear rate for sample together with the deduced viscosity.
range of shear rates, samples in S~ are no longer Newtonian and the viscosity decreases as shear is increased (shear thinning).
Both results can be understood with a simple argument. We expect a non-Newtonian
behaviour when the shear becomes of the order of a characteristic frequency of the sponge [21, 14]. For the most dilute samples in S~, f becomes very large and therefore a non- Newtonian behaviour is seen at accessible shear rate. On the other hand, shear rates larger
than 102 s-I would be needed to reach this regime in Si For example, a crude estimation leads to a critical shear rate of the order of106-103 s-I for the range of dilution accessible in Sj.
The evolution of this viscosity against the water volume fraction ~b is shown in figure 5. One
can observe two distinct regimes. The viscosity varies first weakly with volume fraction for the most dilute samples. Then a sharp increase of the viscosity in the concentrated region is
observed. A possible explanation for this behaviour could be the occurrence of pretransitional
effects announcing a transition towards a solid phase. However, X-ray experiments failed in
observing any fluctuations of this kind. Further experiments are required to clarify this point.
The dilute part is shown in more detail in the insert of figure 5. Considering that the
viscosity is a linear function of
~b for the most dilute samples, the extrapolation to zero membrane volume fraction gives a viscosity higher than the viscosity of the pure solvent (~s~i~~~~ =1.35 mPa.s). A similar result has recently been reported [15] in another sponge
phase system where the solvent is brine.
~
bulk 15
mPas 10
~ o ° .
5 °
0~
0 5 IO 15
o
o o o o . .
Fig. 5. Bulk viscosity of the sponge phases against the water volume fraction. The dilute part of the
curve is given in insert (note that the extrapolation at zero volume fraction is about three times the
solvent viscosity 1.35 mPa.s).
It has been suggested that the structure of the sponge leads to the observed enhanced
viscosity. This explanation would suggest a geometric origin and therefore a somewhat universal value of the enhancement factor [15]. On the other hand, recent theoretical work
gives a non-universal expression of this factor increasing with the bending elastic energy of the
membrane [16]. The enhancement factor is also suggested from a formal study of the
rheological properties of complex interfaces [17].
Flow experiments.
The main purpose of this work is to investigate what happens when the sponge phase is
pushed through pores of size comparable to f. These experiments are described in this section to reveal obstruction effects.
The experiments consist of forcing samples to flow through cylindrical pores. Samples are
first filtered to remove dust particles using membranes having 22001 nominal pore
diameter. We have checked that the sponge phase is not modified by filtration. The
membrane used for preliminary filtration and for measurements are purchased from Millipore (GSWP02500 and PHWP02500), in the form of 25 mm diameter disks with a 150~cm
thickness. Membranes are composed of mixed cellulose ester (nitrate + acetate) ; SEM
picture shows the pores size to be regular. One forces the sponge to penetrate into pores by applying a pressure and then measuring the corresponding flow rate. The differential applied
pressure is determined by U manometers filled with either water or mercury. The flow rate is obtained measuring the time needed for a given volume to go through the filters. Typical
value of the measured flow rate range between 10-3 to 10-2 cm3.s-I
In fact, this experiment leads to a Poiseuille flow. In conventional pipe flow the relation between the differential pressure and the flow rate reads :
D
=
" "~~
ii) 8 ~L
where D is the flow rate (m~.s~~), AP the differential pressure, ~ the viscosity and the ratio
R~/L is
a dimensional characteristic of the pores (R and L are respectively the radius and the
length). If N pores are simultaneously present as in our experiment, the total flow rate simply
reads :
where N is the total number of pores acting in parallel.
This expression can be extended to more complicated cases like the one we discuss in this paper by defining an effective viscosity which eventually becomes flow rate dependent. In any
case, the characterisation of the flow remains the study of the relation between
D and AP. One can also consider a « differential » viscosity defined as : N arR ~d AP
~dif ~ j2)
8 L ~
For convenience, we use in the following the shear associated with D by introducing :
~ 8L
~NarR~ ~
Experimentally, the geometrical factor 8 L/NarR ~ characteristic of the filters is determined by
first measuring the flow for Newtonian samples of known viscosity. We have used dodecane (~
= l.35 mPa.s) and cydohexanol (~
=
68 mPa.s) in order to cover the range of the sponge
phase viscosities.
We first analyse the data for the samples in Si Figure 6a shows the variation of the differential pressure as a function of S for three typical samples along the dilution line. A
blow-up of the same curves is given in figure 6b. Solid lines correspond to the fit using the
model developed in the following. The most concentrated samples exhibit a Newtonian
behaviour with a linear relation between AP and S. The value of the viscosity obtained from the slope is comparable to the bulk one previously measured. However, for more dilute
samples a different behaviour is found with at first a steeper slope and then a crossover towards a regime where the slope is weaker. Since the slope is proportional to the differential
viscosity this indicates a decrease of this viscosity with the shear rate (shear thinning).
A
~~ ~~~ AP
2 ~
X10 ~
"
(3) ~ ~~~
~~
~3~
~ b
(2) (i>
0 2 4 6 8 10 12 0 2 3
5 j
s ~jo~ j~ S x10 s
Fig. 6. (a) Applied pressure drop as a function of the shear rate S (defined in the text) for three different sample in St (~j = 14 fb, ~~ = 5.5 fb, ~~ = 2.7 fb) with filters of pore size 3 0001. The
continuous lines give the fit using the model introduced in the text. (b) Blow-up of the curves to show the low shear (low pressure) part.
Moreover, the initial slope leads to viscosities much higher (1-100 times) than the bulk
viscosity. This effect is more pronounced for the most dilute samples (see Fig. 6b). The
viscosity at small values of shear rate seems to diverge with dilution, I-e- with the characteristic size of the sponge.
Let us first consider the non-Newtonian behaviour. In fact the shear rate obtained in the pores can be estimated from the flow rate and the pore size (Eq, I'). The values obtained are
higher than the one we have investigated in the Couette cell. But the non-Newtonian effect
cannot be only due to a bulk effect. Indeed, for a bulk system, when shear thinning is
expected, the viscosity should vary from the bulk value to a smaller one. In fact, as already stated, the initial slope indicates viscosity much higher than the bulk ones. Thus, the large initial slope should be related to an obstruction phenomena. In agreement with this analysis,
the viscosity retums to a value close to the one found in the bulk for larger shear rates. The fact that the non-Newtonian behaviour found in this experiment may be partially due to bulk effects will be discussed in more details in the following but will be ignored in the simple description of the next paragraph.
Before a quantitative discussion, simple arguments can be given for a qualitative
understanding of the experiments. They are illustrated in figure 7. We study the flow through
pores of a flexible object (the membrane) in solution and we specifically follow the evolution of the system when the object size becomes comparable to the pore diameter (Fig. 7a). The
initial slope, for low pressure, corresponds to the regime where deformation of the sponge is
negligible and consequently the flow of the solvent becomes more and more confined as the
object size reaches a value close to the pore size. This is seen as an enhancement of the initial
slope in figure 6, I,e. of the initial viscosity ~o in the first part of the curves. In this regime,
schematized in figure 7b, obstruction is dominant.
In the second part of the curves, for larger pressure, the flow of the solvent can induce noticeable deformations of the sponge that reduces f (Fig. 7c). This deformation leads in tum to an easier flow of the solvent inside the pores corresponding to a decrease of the viscosity.
For large pressure, an asymptotic value of the differential viscosity ~~ is reached which is no
longer singular. The crossover between the two regimes occurs for a characteristic shear rate S* which is strongly size dependent. Figure 6b shows that S* decreases with dilution. Finally
three input parameters ~o, ~~ and S* are necessary to describe an experimental curve. We
Low shear: High shear:
obstniction defonnat10n
R
@.
- ~ (
(
~
~
L
~
~j
Vo
a b c
Fig. 7. Schematic representations of the flow experiment (a) the sponge is pushed through the pores with a velocity vo. (b) low shear regime; the sponge is schematized with
an object of size f. (c) high shear regime the sponge is deformed when flowing through the pores.
propose in the following a simple model which reproduces the main characteristics of these
curves.
We have also made experiments with the very dilute sample prepared in S~. The aim of this
study is to measure a case where f is larger than the pore size. We give in figure 8 the result for two different pore sizes both smaller than f. Although the shape is similar to the one observed for samples of smaller f, the experiment now suggests the existence of a pressure threshold below which no flow can be observed. When f (larger than the pore size) increases the pressure threshold increases. As we show in the next section the proposed model can also
account for this behaviour.
AP
2
~
.
-F1000i
~~ ~~
~
o°o~'~' ~ , o
. . -F2200i
. "
~~&J
0 5 IO 15 20
5 j
Sx10 s
Fig. 8.- Applied pressure drop as a function of the shear rate S for the sample taken in S~. The result for two different pore sizes (1000 and 2 200 h) is given.
