POZNAŃ UNIVERSITY OF TECHNOLOGY Chemical Technology Faculty Department of Chemical Engineering and
Equipment
Monika Maria Kostrzewa
Studies on the settling of particles in micellar solutions
Badania opadania cząstek w roztworach micelarnych
Doctor of Engineering Thesis Rozprawa doktorska
This thesis was prepared
at the Department of Chemical Engineering and Equipment of Poznan University of Technology,
supervised by Prof. dr hab. Lubomira Broniarz-Press, and
at the Institute of Fluid Mechanics at
Friedrich-Alexander Universität Erlangen-Nürnberg, supervised by Prof. dr hab. Andreas Wierschem
Poznań 2015
i Contents
Abstract ...iv
Symbols and abbreviations ... v
Dimensionless numbers and parameters ...ix
1 INTRODUCTION ... 1
2 AIM OF THE STUDY ... 20
3 RHEOLOGY OF MICELLAR SOLUTIONS ... 21
3.1 Dynamic rheology of micellar solutions ... 21
3.2 Viscosity of micellar solutions ... 26
3.3 Steady shear rheology ... 27
3.4 Time-dependent rheology ... 34
3.5 Rheo-optical properties of micellar solutions ... 35
4 SETTLING OF A SINGLE PARTICLE ... 37
4.1 Settling of a single particle in Newtonian media ... 37
4.2 Settling of a single particle in non-elastic non-Newtonian media ... 45
4.3 Viscoelastic effects on particle settling in non-Newtonian fluids ... 49
4.4 Settling of a single particle in micellar solutions ... 53
5 SETTLING IN MULTI-PARTICLE SYSTEMS ... 56
5.1 Interactions between falling particles in Newtonian fluids ... 56
5.2 Interactions between falling particles in non-Newtonian fluids ... 57
5.3 Sedimentation in Newtonian fluids ... 60
5.4 Sedimentation in non-Newtonian fluids ... 65
6 MATERIALS ... 70
6.1 Chemical compounds used and preparation of the solutions ... 70
6.2 Methods used to characterise the solutions ... 72
6.3 Rheology of the solutions ... 75
6.3.1 Rheology of equimolar CTAB/NaSal solutions ... 75
6.3.2 Rheology of CTAB 10 mM solutions with varying salt content ... 93
6.3.4 Rheology of polymer/surfactant solutions ... 106
6.3.4.1 Micellisation in polymer/surfactant solutions ... 106
6.3.4.2 Rheology of polyelectrolyte/surfactant solutions ... 109
7 RESULTS AND DISCUSSION ... 121
7.1. Measuring setup ... 121
7.2 Settling of particles in CTAB/NaSal solutions ... 124
7.2.1 Settling of a single particle in equimolar CTAB/NaSal solutiuons ... 124
ii
7.2.2 Interactions between falling particles in equimolar CTAB/NaSal
solutions ... 140
7.2.3 Settling of a single particle in varying salt content CTAB solutions . 159 7.2.4 Interactions between falling particles in varying salt content CTAB solutions ... 164
7.3 Settling of particles in polymer/surfactant solutions ... 180
7.3.1 Settling of a single particle in polymer/surfactant solutions ... 180
7.3.2 Sedimentation in polymer/surfactant solutions ... 192
8 CONCLUSIONS ... 198
References ... 200
Streszczenie ... 215
List of Figures ... 227
List of Tables ... 233
Appendix ... 234
Curriculum Vitae ... 258
List of Publications ... 259
iii Acknowledgments
I especially wish to express my deep appreciation to my supervisors, Prof. dr hab. Lubomira Broniarz-Press and
Prof. dr hab. Andreas Wierschem,
for their encouragement, support and guidance during my research.
Your expertise and invaluable comments helped me greatly in completing this dissertation.
My acknowledgments also go directly to the
Deans of the Chemical Technology Faculty at Poznan University of Technology, to Prof. dr hab. Andrzej Olszanowski and Prof. dr Eng hab. Krzysztof Alejski,
for giving me the great opportunity to continue my research while on a foreign scholarship.
Many thanks and much gratitude go to the
Head of the Institute of Fluid Dynamics at Friedrich-Alexander Universität, Prof. dr Eng hab. Antonio Delgado,
who granted me this opportunity to join the Institute as an assistant researcher.
I also direct my acknowledgments to the BAYHOST foundation for their financial support during the scholarship.
To all who believed that this day would come
iv Abstract
In this thesis, the influence was studied of additives, such as the strongly binding counter ion NaSal, polyelectrolyte NaCMC and non-ionic polymer PEO, in surfactant solutions on particles settling in micellar solutions. The solutions chosen here are shear thinning viscoelastic fluids. It has been observed that particles falling in semi-dilute CTAB/NaSal systems behave differently than in polymer liquids.
Equimolar CTAB/NaSal 20/20-80/80 mM solutions are characterised by the shear stress plateau, which is indicated by complex steady-shear rheology due to the coexistence of bands with different micellar structures (shear banding). Based on the rheo-optical measurements conducted for the CTAB/NaSal 80/80 mM solution, it was shown that turbidity emergence and different scattering patterns take place in connection with shear banding. Here the shear flow visualisation showed that the shear-banded structures are time-dependent and that fluctuations appear at the boundary of the bands. On the other hand, the shear stress plateau has not been stated in CTAB 10mM/NaSal 10-80mM systems. Nonetheless, rich time-dependent rheology is ascribed to the rheo-chaos phenomenon, suggesting transformation of the micellar network due to the salt excess and the shear flow. To combine the rheological properties with particle motion, an effective shear rate, defined as the ratio of particle velocity and its diameter, has been introduced. It was found that the single particle falling in equimolar CTAB/NaSal solutions starts to move unsteadily at an effective shear rate corresponding to the shear stress plateau. The onset of flow instability and drag enhancement are met at a concentration of 50/50 mM, at which the minimum of the relaxation time is stated. It was concluded that structural time scales determine particle oscillations in equimolar solutions. Additionally, in equimolar CTAB/NaSal solutions the relationship between particle velocity and micellar network size has been shown. The salt excess in CTAB 10 mM solutions causes an increase in particle velocity. Again, drag enhancement was found at the minimum of the relaxation time. Independently of the salt concentration, the particle falling in CTAB 10mM/NaSal solutions oscillates with similar frequencies as the shear rate fluctuates, which emphasises structural changes due to the shear flow.
Studies on interactions between particles falling in CTAB/NaSal solutions showed that the concentration and initial separation distance are the main factors influencing the behaviour of the falling particles. Again, the micellar structure determines the spheres’ behaviour, which highlighted the conclusion that was made for a single particle. In contrast to polymer liquids, in CTAB/NaSal solutions it has been found that the trailing spheres oscillate or retard. A particle settling in polymer/surfactant solutions reaches terminal velocity. Although the micellar network architecture in polymer/surfactants is quite different from that in CTAB/NaSal solutions, it was concluded that particle velocity is also determined by the micellar structure. The prediction of particle velocity in surfactant/polymer solutions using equivalent parameters modified by the Ostwald-de Waele model gave satisfactory results. For diluted suspensions settling in NaCMC/surfactant solutions, it has been shown that the Richardson and Zaki model is applicable when anticipating that the shear thinning properties of the solutions used are taken into account.
v Symbols and abbreviations used in this study Symbols
A – area [m2]
avd specific area [m2]
a0 – cross-sectional area [m2]
C – configuration tensor C in the deformed material [-]
CD – drag coefficient [-]
CM – molar concentration [mol/dm3]
CMsalt – molar concentration of salt [mol/dm3]
CMsurf – molar concentration of surfactant [mol/dm3]
Cp – packing parameter [-]
D – diameter of container [m]
d – diameter [m]
de – equivalent diameter [m]
dp – diameter of the particle [m]
dr – reduced diameter [-]
F – force [N]
Fb – buoyancy force [N]
FC – FENE spring force [N]
FD – drag force [N]
Fg – gravitational force [N]
FP – force acting on the particle [N]
f – frequency [Hz]
f(d/D) – Faxen’s series [-]
G – Hooke constant [ms2/kg]
G’ – storage modulus [Pa]
G’’ – loss modulus [Pa]
"
Gmin – minimum of loss modulus [Pa]
0
GN – plateau of storage modulus [Pa]
g – gravitational acceleration [m/s2]
h – height [m]
H – spring constant in Eq. (33) [-]
h0 – initial drop height [m]
Iapp – light intensity at shear rate [-]
I0 – reference light intensity [-]
K – drag correction factor [-]
K’ – constant in Eq. (149) [-]
K’’ – constant in Eq. (153) [-]
KN – wall correction factor [-]
KU – viscometer constant [mm2/s2]
K/KN – normalised drag correction factor [-]
k – fluid consistency coefficient [Pa·sn]
k0 – Boltzmann constant [J/K]
L – contour length [m]
lc – alkyl chain length [m]
le – entanglement length [m]
lp – persistence length [m]
M – molecular mass of the surfactant [kg/kmol]
Mh molecular mass of the hydrophilic part [kg/kmol]
vi
m – mass [kg]
N1 – first normal stress difference [Pa]
N2 – second normal stress difference [Pa]
n – flow index [-]
Nc – number of carbon elements in alkyl chain [-]
n’ – coefficient in Eq. (130) [-]
n’’ – coefficient in Eq. (131) [-]
Δn’ – flow birefringence coefficient [-]
p – hydrostatic pressure [Pa]
Δp – pressure drop [Pa]
R – radius [m]
rcs – cross-sectional radius [m]
Q – connector vector between the beads [-]
Q2 – extension of the connector vector [-]
2
Q0 – maximum of possible spring extension [-]
s – distance between the particles [m]
s* – critical separation distance [m]
s/d – dimensionless distance between the particles [-]
T temperature [K]
t – time [s]
u – particle–side–migration velocity [m/s]
w0 – fluidisation velocity [m/s]
V – volume [m3]
VH – volume occupied by the hydrophobic group [m3]
Vbed – volume of bed [m3]
Vf – fluid volume [m3]
Vsus – suspension volume [m3]
v – settling velocity [m/s]
vl – leading sphere velocity [m/s]
vp – particle velocity in Newtonian fluids [m/s]
vr – dimensionless reduced velocity [m/s]
vside – velocity of side-by-side falling particles [m/s]
vt – terminal velocity in non-Newtonian fluids [m/s]
vt – trailing sphere velocity [m/s]
vtandem – long body velocity [m/s]
v0 sphere velocity at the fluid surface [m/s]
v* – dimensionless velocity [-]
w0 velocity through the bed [m/s]
X – correction factor [-]
X(n) – deviation factor for power law fluids [-]
Xwall – drag correction factor in Eq. (108) [-]
X∞ – drag correction factor in unbounded flow in Eq. (108) [-]
x – coordinate [-]
y – coordinate [-]
Z – exponent in Richardson and Zaki model [-]
zp – penetration depth [-]
α – angle [°]
α1 – fraction in band [-]
β – mobility parameter [-]
γ – deformation [%]
vii
γ0 – amplitude of the deformation [%]
– shear rate [s-1]
app surface-averaged particle shear rate [s-1]
eff – effective shear rate [s-1]
max – maximal shear rate [s-1]
– average shear rate [s-1]
1 – shear rate in the band [s-1]
2 – shear rate in the band [s-1]
1
crit – critical shear rate for the onset of the shear stress plateau [s-1]
2
crit – critical shear rate for the end of the shear stress plateau [s-1]
δ – phase shift [-]
δe – equivalent linear diameter [m]
ε – porosity [m/s]
ε0 – initial porosity [m/s]
η – viscosity [Pa·s]
ηa – apparent viscosity [Pa·s]
ηc – Cassonian viscosity [Pa∙s]
ηeff – effective viscosity [Pa∙s]
ηpl – plastic viscosity [Pa·s]
ηs – solvent viscosity [Pa·s]
ηsus – suspension viscosity [Pa·s]
ηred – reduced viscosity [Pa·s]
η0 – zero-shear rate viscosity [Pa·s]
η∞ – infinity viscosity [Pa·s]
η* – complex viscosity [Pa·s]
κ – interaction parameter [-]
Λ – Carreau number [s]
λ – relaxation time [s]
λbr – breakdown time [s]
λc – time constant in Eq. (7) [s]
λrep – reptation time [s]
ν – kinematic viscosity [mm2/s]
ζ – mesh size of the micellar network [m]
ρf – fluid density [kg/m3]
ρs – solid density [kg/m3]
σ – surface tension [mN/m]
σxx, σyy, σzz – dynamic components of the stress [Pa]
τ – shear stress [Pa]
τf – yield stress [Pa]
τplateau – plateau of shear stress [Pa]
τxx, τyy, τzz – flow induced normal stress components [Pa]
τxy – shear stress in simple shear flow [Pa]
τ1 – fast relaxation time [s]
τ2 – slow relaxation time [s]
ϕ – volume fraction [-]
ψ – sphericity [-]
ω – angular frequency [rad/s]
ωe – equivalent velocity [m/s]
viii Abbreviations
CAC – critical aggregation concentration CMC1 – first critical micellisation concentration CMC2 – second critical micellisation concentration CpyCl – cetylpyridinium chloride
CTAB – hexadecyltrimethylammonium bromide CTAC – hexadecyltrimethylammonium chloride CTAT – cetyltrimethylammonium tosylate
DR – drag reduction
ER – electrorheological fluids
FENE – finitely extendable nonlinear elastic chain model HEC – hydroxyethyl cellulose
HLB – hydrophilic – lipophilic balance HTAC – hexadecyltrimethylammonium chloride MR – magnetorheological fluids
NaBen – sodium benzoate NaCl – sodium chloride
NaCMC – sodium carboxymethyl cellulose NaSal – sodium salicylate
O/W – oil/water
hmPAM – hydrophobically modified polyacryloamid PEO – poly(ethylene) oxide
PIV – particle image velocimetry PMSA – p – methyl acrylic acid PSP – polymer saturation point PTV – particle tracking velocimetry
R – alkyl chain
RMS – root error mean square ROI – region of interest
SALS – small angle light scattering
SANS – small angle neutron light scattering SDS – sodium dodecyl sulphate
SIS – shear induced state SOR – stress optical rule
TMAH – tetramethylammonium hydroxide TTAB – tetradecyltrimethylamonium bromide
W/O – water/oil
WLM – wormlike micelle VES – viscoelastic surfactants
X – counter ion
ix Dimensionless numbers and parameters
Ar
2 3
f c f
s g
Ar d
Archimedes number for Newtonian fluids
Arp 2
2 2 2 2
3 4
k g Ar d
n n n
p
Archimedes number for power law fluids
Fe 0
2 0
GN
Fe v elastic Froude number
Fr gd
Fr v
2
0 Froude number
Gr 0
N
e G
Gr gd elastic Grashof number
Re
Revd Reynolds number for Newtonian fluids
Rep
k d Re v
n 2
2
Reynolds number for power law fluids
Re0
0
0
d
Re vt Reynolds number for the Carreau model
Wi d
Wiv Weissenberg number
Γ
f f s
density number
Λ
p t c
d
v
2
Carreau number
x
1 1 INTRODUCTION
Surface-active agents (surfactants) are organic compounds which are of great scientific interest as they are used in many industrial branches. The fundamental property of surfactants is their ability to lower the surface tension of solutions and to accumulate at surfaces and the interface [1-4]. In general, a single surfactant molecule has an amphiphilic structure owing to its composition of a hydrophilic part (called the “head”) and a hydrophobic part (called the “chain”), as is presented in Figure 1.
Based on this definition, other more complex surfactants may arise, e.g. gemini surfactants have more than one chain linked by a “spacer”, or polymeric surfactants that are interlinked [1-4]. According to their origin, surfactants can be classified into natural and synthetic chemical compounds. Natural surfactants may be obtained from either renewable (e.g. plant oils, mono/polysaccharides, vegetable/animal fat) or non- renewable sources (e.g. crude oil, hard coil, petroleum). When considering the influence on the environment, it is possible to divide surfactants into biodegradable and chemo-degradable compounds [1-4].
Figure 1. Representation of a surfactant molecule [1]
The hydrophilic part is characterised by its affinity to aqueous phases and other polar solutions. The polar part of surfactant molecules is often a derivative of water- soluble organic salts, such as deprotonated acids (e.g. carboxylate group R−COO–, sulfonate group R−SO3–) or amine groups (−NH2; −NHR; −NR2). Moreover, groups which cannot build salts and are not soluble in water may occur in the surfactant molecule (e.g. alcohol –OH or ester groups R–COO–R, where R stands for the chain substituents) [1]. On the other hand, the hydrophobic part is characterised by its affinity to non-polar solutions, and they can be composed of an alkyl chain or a conjugated aromatic chain. Typically, the alkyl chain consists of 8 to 18 carbon atoms which may be linear, aliphatic or branched [1, 2]. Structural aspects such as
2
degree of branching, alkyl chain length and location of the polar group in surfactant molecules are the key parameters of these molecules dictating their physicochemical properties.
Here, the general classification of surfactants is based on the charge of the hydrophilic part, hence ionic surfactants (anionic, cationic and zwitterionic) and non- ionic surfactants are singled out as the two main subgroups [1-4].
Since anionic surfactants are used as detergency agents, they comprise 60% of general worldwide surfactant consumption [3, 4]. The most commonly used hydrophilic groups in the anionic surfactant’s molecule are carboxylates CnH2n+1COO–X, sulfates CnH2n+1OSO3–X, sulfonates CnH2n+1SO3–X and phosphates CnH2n+1OPO(OH)O–X, here X are the counter ions such as Na+, K+, Ca+, Mg+ [4].
The optimal detergency is obtained at alkyl chain lengths with carbon atoms n =12–
18 [4]. The most representative of the carboxylate- and sulfate-based surfactants are soaps, e.g. C17H35COONa and sodium dodecyl salicylate (SDS), respectively.
Sulfonate surfactants can be obtained by sulfonation of, e.g. lignin, low-cost hydrocarbon and petroleum fractions; for instance, paraffin sulfonate is commonly used in detergents, emulsifiers, wetting agents and dispersants [4]. Phosphate- containing surfactants are formed from fatty alcohols and alcohol ethoxylates with POCl3 and P4O10. Owing to their anticorrosive properties, phosphate anionic surfactants are used in the metal industry [4].
Cationic surfactants are amine or quaternary ammonium compounds with the general formula of R’R”R’’’R””N+X-, here X corresponds to the chloride ion.
Besides the nitrogen-based cationic surfactants, phosporan, sulfate and sulfoxonium cationic surfactants are also known [4]. Since cationic surfactants can adsorb at negatively charged surfaces, they are used as anticorrosive agents, flotation collectors for mineral ores, dispersants for inorganic pigments and antistatic agents.
Zwitterionic or amphoteric surfactants contain two charged groups with different signs, e.g. N-alkyl derivatives of betaine (CH2)2NCH2COOH) or glycerine (NH2CH2COOH), therefore their properties depend on the pH of the solution. Due to the presence of quaternary ammonium groups, the zwitterionics can behave as anionic or cationic surfactants. Zwitterionic surfactants have good dermatological properties, therefore they are used in personal care products [4].
3
Non-ionic surfactants are the second largest class of surfactants. Their polar groups are based on ethylene oxide, therefore these compounds are also known as ethoxylated surfactants; examples are sorbitan esters, e.g. fatty acid ester (the trade name is Span), and their ethoxylated products known as Tween, which are used in the food and drug industry. Another important group of non-ionic surfactants are fatty alcohol ethoxylates which are used as oil-water stabilisers [4]. An important parameter for determining non-ionic surfactant application is the hydrophilic- lipophilic balance (HLB). The term ‘HLB’, as proposed by Griffin, is based on the following equation [1, 5]:
M
HLB20Mh (1)
The hydrophilic-lipophilic balance number is expressed as the ratio of the molecular mass of the hydrophilic part Mh to the molecular mass of the surfactant molecule M [1]. Different values and examples of their application are found in Table 1.
Table 1. Application of surfactants based on HLB values [1,5]
Value of HLB Application 1.5 - 3 antifoaming agents
3 - 6 water/oil emulsifiers
7 - 9 wetting agents
8 - 18 oil/water emulsifiers
13 - 15 detergents
15 - 20 solubiliser
Due to the surfactants’ ability to reduce surface tension and surface elasticity, they can be used as foaming and antifoaming agents. Non-ionics with a branching- chain still show surface activity, but they prevent foam formation, which can disturb some industrial processes [3]. Two immiscible pure liquids, such as water and oil, can form two types of emulsions, water/oil and oil/water, by the addition of oil- soluble or water-soluble surfactants such as Span or Tween, respectively [3].
Surfactants with a short alkyl chain length (nC = 8) or micelles with a short lifetime are used, e.g. in powder technologies as wetting agents which help liquid penetration into the pores or channels of solid materials. Nevertheless, a good wetting agent is not necessarily a good detergent. Surfactants with 12-16 carbon atoms in the alkyl chain, due to the ability of adsorption at the interfaces and a simultaneous decrease in
4
surface tension, are widely used as detergents. Considering surfactants as good detergents means they have to be good wetting and solubising agents at the same time since dirt needs to be removed into the washing fluid and then be solubilised [4]. Ethoxylated surfactants, due to a further increase in the alkyl chain length, become solubiliser agents which are able to dissolve liquids but also solids and gases [3]. Davies extended Griffin’s HLB number concept by introducing hydrophilic and lipophilic group numbers [2, 6]:
HLB = 7+∑(hydrophilic group number) + ∑(lipophilic group number) (2) Here the sign of the hydrophilic group number has a negative value, while for the lipophilic group it is positive. This method allows to classify lipophilic (HLB < 9) and hydrophilic (HLB >9) non-ionic surfactants [7]. Another approach to polyethoxylated surfactants was proposed in [8]. The HLB number is calculated with an effective chain length which is based on the critical micelle concentration [8].
Although the HLB value is commonly accepted, it cannot be used as a universal parameter for the selection of surfactants since it does not take into account the role of the temperature and addition electrolytes on the surfactant’s final properties [2].
A characteristic property of surfactants is their ability to arrange into complex structures, called micelles [1-4]. Here both the surfactant concentration and the competition of electrostatic and hydrophobic interactions are the driving forces for their formation [9]. The formation of such structures as a function of the surfactant concentration is presented in Figure 2. At a low surfactant concentration, single surfactant molecules (monomers) aggregate at the air–solvent surface, in which the hydrophilic part is oriented into the water phase. An increase in the surfactant concentration results in saturation of the interfacial layer with the surfactant molecules. A further increase of the concentration drives to spontaneous agglomeration of molecules into micelles. Such a transition takes place at a concentration called the ‘critical micelle concentration’ – (CMC1) [1-4]. Each surfactant exhibits a characteristic value of the critical micelle concentration CMC1
at a given temperature [1-4, 7]. In micellar agglomerates the hydrophobic parts form the core of the micelle, while the hydrophilic groups form the shell in contact with the aqueous phase.
5
Figure 2. Arrangement of surfactants into micelles with an increasing concentration [2]
Above CMC1, the micelles remain in thermo-dynamical equilibrium with individual surfactant molecules [4]. The fast, intensive exchange between micelles, monomers and the surrounding bulk phase takes places during the characteristic relaxation time τ1 [10]. A second relaxation process is associated with micelle breakdown and formation and thus with the lifetime of the micelles [10]. Its time scale τ2 is much longer than τ1.
The most common technique for determining CMC1 is by measuring surface tension, but other techniques can also be applied, e.g. spectroscopic methods, voltammetry, scattering techniques, and calorimetry. The formation of micelles is indicated by the break-up point in the function of the physicochemical properties versus the surfactant concentration [7]. Several factors can influence CMC1, as listed in Table 2.
Table 2. Main factors influencing the CMC1 value [2-4, 7]
CMC1 decrease with: CMC1 increase with:
– an increase of the hydrophobic chain length
– replacement of the hydrocarbon-based hydrophobic group by the fluorocarbon- based group
– addition of electrolytes, alcohols and organic counter ions into ionic surfactant solutions
– increase of chain length in the non-ionic surfactant molecule
– presence of double bonding and branching in the hydrophobic tail
– introduction of polar groups into the hydrophobic tail
– number and central location of hydrophilic groups
– use of highly polar solvents
The main factors that influence CMC1 are the surfactant structure, temperature and presence of electrolytes and of organic compounds [1-4]. Depending on the charge, it
6
has been noted that non-ionic surfactants with the same alkyl chain length possess a CMC1 of at least two order of magnitude lower than the ionic surfactants, whereas cationic surfactants are characterised by a higher CMC1 than the anionic ones [2-4].
Temperature impact on CMC1 is quite complex. Generally, the temperature increase reduces the CMC1 value; however, in extreme cases, at much higher temperatures the CMC1 values can increase [3]. Together with the surfactant concentration, the temperature determines the phase-behaviour of a surfactant solution. At a certain temperature and concentration of CMC1, a characteristic Krafft point is stated which corresponds to the maximum solubility of monomers [1-4]. The decrease in temperature (below the Krafft point) results in the solubility reduction and dissolution of micelles, while above this point the solubility increases and the micelle formation is favourable. The location of the Krafft point changes along with the alkyl chain length and the presence of electrolytes [1-4].
The morphology of the micellar shape depends on the character of the surfactant [1-4, 11]. Israelachvili [11] classified the micelle shape by using a packing parameter as given:
a0
l C V
c H
p (2)
where VH is the volume occupied by the hydrophobic groups in the core, lc is the alkyl chain length and a0 expresses the cross-sectional area of the occupied hydrophilic parts on the micelle surface. Depending on the packing parameter as listed in Table 3, various structures are expected.
Table 3. Packing parameter and the respective shapes of the surfactant aggregates [11]
Packing parameter Cp Structures formed
< 1/3 spherical micelles
1/3 - 1/2 cylindrical micelles
1/2 - 1 flexible bilayers, vesicles
~ 1 planar bilayer
> 1 inverted micelles
The simplest-formed agglomerate is a spherical micelle where the hydrophobic parts form the micelle core and the hydrophilic tails are directed to the polar phase (Figure 3a). This agglomerate consists of 50-100 monomers with a radius similar to the length of the extended hydrocarbon chain and with a size of about 50 Å [3, 11].
7
The spherical micelles are formed by the ionic surfactants with a large head group, e.g. sodium dodecyl sulfate (SDS) [1-4, 11]. Yet the composition of the spherical micelles created by the non-ionic surfactants may be different [3]. Around the core, the head groups and bounded counter ions form a Stern layer. The surrounded free counter ions form a Gouy-Chapman layer which is extended into the aqueous phase.
On the other hand, the outer region of the non-ionic surfactant’s micelle does not contain the counter ions and is built by the coils of the hydrated polyoxoethylene groups [3].
a) b)
c) d)
e) f)
Figure 3. Different shapes of micelles [1, 3, 4, 12]:
a – spherical, b – globular, c – rod-like, d – reverse, e – lamellar, f – vesicle
Globular micelles (Figure 3b) are micelles with a radius that is larger than the extended length of the surfactant chain. This structure is expected when the spherical micelle cannot pack more surfactant molecules [12] or due to the solubisation of hydrocarbon chains in rod-like micelles (Figure 3c) [13]. The rod-like micelle has a cylindrical part and spherical endcaps with different diameters [12].
Surfactants with small head groups and very large and bulky double hydrophobic tails form different structures depending on the solvent polarity [3]. In
8
non-polar media, reverse micelles form due to the dipole-dipole interactions between the hydrophilic groups (Figure 3d). In polar media, lamellar micelles appear and form vesicles at high surfactant concentrations (Figure 3e and f) [3, 14]. The addition of non-polar substances into aqueous solutions may also result in a conversion of lamellar micelles into reverse micelles or an inverted lamellar structure [3].
Depending on the temperature, surfactant concentration and additives into the liquid phase, the spherical micelles may change their shape. Elongated, cylindrical micellar structures are formed by surfactants with close-packed small head groups and short, bulky hydrophobic chains (Figure 4a) [3]. With an increase in the surfactant concentration the transition from cylindrical to wormlike micelles is stated by the second micelle concentration – CMC2 [15]. Surfactants with a quaternary ammonium ion form threadlike complex structures upon the addition of a structure- forming substance such as salt (Figure 4b) [16-19]. Inorganic salts, such as NaCl or KBr, promote an increase in the micelle’s length due to the screening of electrostatic repulsions between the charged head groups and ions [16, 20]. A simple ion is adsorbed only at the micelle interface, thus resulting in a decrease of the area [16, 19]. The addition of an organic salt (called hydrotropes) leads to faster and more efficient growth of the complex structures [16-19]. In the case of an organic salt, an aromatic ring is located between the head groups indicating an increase of the micelle volume and lowering electrostatic repulsive forces between the head groups [19]. A wormlike micelle is characterised by the persistence length lp, contour length L and the cross-sectional radius rcs [16]. The persistence length pertains to the rigid part of the wormlike micelles and is about 300-600 Å. The contour length L increases exponentially at higher concentrations. The cross-sectional radius rcs is independent of the concentration, similar to the spherical micelles and varies from 20 to 25 Å [3, 16].
At low concentrations, wormlike micelles do not interact with one another, while in a semi-dilute regime the micelles may form a linear, branched or multi-connected network [17, 18, 20-22]. The micellar network is fully entangled and the distance between the entanglement points is characterised by mesh size ζ, which decreases in the concentrated range [17]. The transition from a linear to a branched micelle has been reported due to salt excess [20, 21, 23, 24]. The free endcaps may join to the
9
micelle chain and form a branching point [23]; the formation of the branching point will be discussed in Section 3.1. At the branching point it has been suggested that the micellar chains may slide, thus indicating the temporary character of this binding [23, 24]. An example of a branched micelle is presented in Figure 4c.
a)
b)
c)
Figure 4. Examples of cylindrical (a), wormlike (b), and branched (c) micelles [2, 15, 23]
The addition of a polyelectrolyte into wormlike micellar solutions leads to the formation of threadlike hybrid micelles [25]. The charged part of the polyelectrolyte is located between the head groups, and the backbones are placed close to the micelle interior [25]. Most of the experimental studies on polymer/micelle interactions concerned the polyelectrolyte and oppositely charged surfactant, or the association of a non-ionic polymer with the micelles. When considering the interactions between the polyelectrolyte and surfactants, they are typically characterised by the critical aggregation concentration – CAC. The CAC has lower values than CMC1 and depends on the polyelectrolyte concentration [2, 26]. It has been reported that an increase in the polyelectrolyte concentration shifts the CAC to higher values [26].
Although beyond the CAC the surface tension barely changes with the surfactant concentration increase, nonetheless a continuous saturation of polymer chains with
branching point
10
the micelles takes place up to a certain concentration called the polymer saturation point (PSP) [27]. Beyond the PSP the polymer-surfactant and free micelles coexist in an equilibrium state [2, 27]. The shape of the polyelectrolyte/surfactant micelles depends on several factors, such as the surfactants’ hydrophobicity, the strength of the mutual electrostatic and hydrophobic interactions [9] and the molar mass of the polyelectrolyte [26, 28]. The dominance of hydrophobic interaction causes the micelles to be surrounded by a polyelectrolyte chain. On the other hand, strong electrostatic interactions lead to stretching of the polyelectrolyte chain and the formation of bottlebrush structures [9]. An increase in the molar mass causes lengthening of the polymer chains, which are then more able to gradually surround the micelle [25-28].
In oppositely charged polyelectrolyte/surfactant systems, electrostatic interactions are dominant [26]. The addition of a cationic surfactant into the anionic polyelectrolyte solutions causes charge neutralisation and, sometimes, just as in NaCMC/CTAB systems, precipitation may take place at ratios of cationic to anionic ions above unity [26]. Nevertheless, the precipitation can be counter-balanced by the addition of salt, however, the binding affinity between the polyelectrolyte and oppositely charged surfactants in diluted systems is reduced [29-31]. Above the CAC the polymer chain may collapse, which results in significant viscosity reduction. In some cases at relatively high surfactant concentration the solution viscosity is close to that of the solvent [29]. Hydrophobic interactions between the anionic polyelectrolyte and the anionic surfactant are more dominant than the electrostatic ones [30, 32]. The CAC in these systems is lower than CMC1
and is independent of the polymer concentration [33]. The micelles in these systems are different from those formed by pure surfactants; the micelles attach to the polymer backbone [33, 34].
The interactions between non-ionic polymers and surfactants strongly depend on the charge of the head group of the surfactant and are more pronounced in polymer/anionic surfactant solutions. As is presented in [35], at the CAC the anionic surfactant micelles are bound to the polymer chain, and free molecules are present in the solutions. The micelles formed on the polymer chain do not change their coil.
The polymer chains surround the micelles, whereas some of the chain segments are
11
located at the hydrocarbon/water interface [36, 37]. As was mentioned above, the interaction between the cationic surfactants and non-ionic polymers is weak [36, 38]. As was reported by Brackman and Engberts [36], the addition of a non-ionic polymer does not influence the CMC1 value and the micelle’s shape. Only an increase in viscosity and viscoelasticity was observed [39]. Nevertheless, interactions between the cationic surfactant/polymer may become stronger due to an increase in the hydrocarbon chain length in the surfactant molecule [38] or by the addition of electrolytes [39].
At a certain concentration of micelles the packing of micelles leads to a particular arrangement known as liquid crystal [3]. This geometric arrangement strongly depends on the shape of an individual single micelle. Hence, as is presented in Figure 5, spherical micelles form a cubic liquid crystal (Figure 5a) and lamellar liquid crystals are created by lamellar micelles (Figure 5b). The nematic phase is typical of cylindrical micelles at a surfactant concentration of 20-30% wt. or may form due to shear flow [17]. A further concentration increase (up to about 40%) drives to the formation of a hexagonal liquid crystal as is presented in Figure 5c [3, 17]. These structures are fully arranged and the micellar solutions behave like a strong elastic gel with a yield stress [17].
a) b) c)
Figure 5. Geometric arrangement of micelles in liquid crystal phase [3, 17]:
a – cubic-arranged spherical micelles, b – lamellar liquid crystal, c – hexagonal-packed cylindrical micelles
The different shapes and complex structures of the surfactant aggregates determine the rheological properties of the micellar solutions. With rheological measurements it is possible to study the complex flow behaviour of micellar solutions. To this end, the most important terms in rheology will now be introduced.
12
Considering the steady unidirectional flow, a fluid located between two parallel plates with an area A under the force applied F experiences a shear stress τxy, as described by equation (3), called Newton’s law [40]:
dy
dv A
F
xy (3)
In Newtonian fluids, viscosity remains constant at a constant temperature. The flow curve is a straight line passing the origin of the coordinate system [40]. On the other hand, the viscosity of non-Newtonian fluids can change along with the flow conditions and geometry, as given by equation (4):
f
(4)
Non-Newtonian behaviour is classified into the following subgroups [40-42]:
– time-independent, e.g. pseudoplastic (shear thinning), dilatant (shear thickening), yield stress fluids with magneto-(MR) and electro-rheological (ER) fluids as yield stress controlled fluids,
– time-dependent, e.g. thixotropic, rheopexic;
– viscoelastic;
– rheo-optical.
The flow curves for the rheostable and time-dependent fluids are presented in Figure 6. When considering the time-independent fluid, two mechanisms are pointed out; shear thickening and thinning [40]. For the shear thickening fluids the viscosity increases with the shear rate. Most often such behaviour occurs in concentrated suspensions (e.g. corn flour in water, sand in water). Here the fluid plays the role of a lubricant. At low shear rates the inertial friction is maximal, and along with the shear rate increase the particle-particle interactions become dominant and, finally, at high shear rates a cluster formation is responsible for the shear thickening behaviour [40].
In the case of shear thinning fluids (e.g. polymer solutions or melts), viscosity decreases along with the shear rate due to the adaptation of fluid to the flow conditions. At rest and at low shear rates the molecules are randomly oriented and the inertial friction is maximal because Brownian motions dominate. In these conditions the fluid behaves Newtonian because the molecules can resist the deformation, therefore they remain undeformed or entangled. At moderate shear rates the
13
molecules like, e.g. entangled polymer coils, and can no longer resist the deformation. Therefore the polymer coils start to align into the flow direction and to deform along the streamlines, hence the viscosity decreases. At high shear rates a second Newtonian regime is pointed out, indicating complete elongation or destruction of, e.g. the polymer chains [40].
a) b)
Figure 6. Flow curves of chosen non-Newtonian fluids [40-42]:
a – time-independent, b – time-dependent fluids
Yield stress fluids are the third group of time-independent fluids. At rest the substance behaves like a solid with an efficient rigidity. If the stress applied is greater than the yield stress τf the material structure breaks down and the medium starts to flow. Therefore the flow curve does not pass the origin of the coordinate system but starts at τf. One of the interesting classes of yield stress fluids are electro- and magneto-rheological fluids, sometimes called “smart fluids”. Electrorheology and magnetorheology refer to the change of rheological properties of the solutions due to an electric or magnetic field [41]. These fluids are usually dispersions which rearrange in an imposed field. By placing, e.g. an MR fluid in a magnetic field, the particle clusters appear and the viscosity increases. ER fluids behave in a similar way [41, 43]. MR and ER fluids are commonly used in the automotive industry, however, MR fluids have some advantages over ER fluids. The MR fluids cause larger rheological changes than the ER fluids and are less sensitive to moisture and contaminants [43]. In order to avoid sedimentation of the ferromagnetic particles in
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MR fluids, surfactants, e.g. tetramethylammonium hydroxide (TMAH), are used.
Such solutions, however, may not be applicable in ER fluid [43].
Table 4. Rheological models for time-independent fluids [40-42]
Models Relation Remarks
Ostwald-de Waele
[44]
1
kn
n [-] and k [Pa∙sn] parameters correspond to the flow index and fluid consistency coefficient, respectively. For n < 1 shear thinning and for n > 1 shear thickening behaviour is indicated.
(5)
Ellis model [45]
1
2 1
0
1
/
α[-] expresses the shear thinning degree, τ1/2[Pa] is the shear stress value which corresponds to half shear stress at zero shear viscosity
(6)
Carreau-Yassuda model
[46, 47] 0 0
1 2
21
c n
η0 corresponds to zero shear viscosity, η∞ is infinity viscosity, λc is the time constant, n indicates the flow index
(7)
Bingham model
[48] f pl τf [Pa] is the yield stress and ηpl [Pa∙s] is the plastic viscosity
(8)
Herschel-Bulkley model [49]
n
f k
τf is the yield stress and n and k have the same physical meaning as in Eq. (5) (9)
Casson model
[50] n
c nf n
1 1
1
n is an exponent, for n = 1 Eq. (10) simplifies to the Bingham model, for n=2 the Eq. (10) describes Casson fluids
(10)
The viscosity curves of the time-independent fluids are described by the frequently used relations as listed in Table 4. The Ostwald-de Waele model (5) may describe the viscosity curve in the shear thinning or thickening regime. The Ellis model (6) is appropriate for shear thinning fluids with the first Newtonian regime.
The Carreau-Yassuda model (7) takes into account the presence of regimes with the zero shear viscosity η0 at low shear rates and high shear rate infinity viscosity η∞. When considering yield stress fluids, the Bingham model describes the linear increase of shear stress above τf (8), while the Herschel-Bulkley (9) describes shear thinning or thickening behaviour above the yield stress [40]. The rheological
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properties of some biological materials, such as blood or food products, are described by the Casson model (10) [41].
The second group of non-Newtonian fluids shows an instantaneous and time- dependent response to stress due to changes in the material structure [40-42]. As is presented in Figure 6b, thixotropy and rheopexy are singled out as two time- dependent behaviours. For the thixotropic materials the decrease in apparent viscosity over time at a fixed shear rate is observed due to the slow breakdown of the structure. This change may be reversible; the structure is rebuilt after cessation. On the other hand, rheopectic fluids build structures by the shear flow. Such behaviour is observed in high particle concentration systems [40-42].
Many phenomena cannot be described qualitatively by the viscosity function and elastic properties have to be taken into consideration. Most time-independent fluids such as polymer solutions under deformation show intermediate responses between elastic and viscous; they are viscoelastic. Viscoelastic materials are able to store and recover the energy during the shear flow. To describe the viscoelastic behaviour, a dashpot immersed in the viscous fluid (Newton element) and an elastic spring (Hooke element) connected to each other are used.
a) b)
Figure 7. Representation of linear viscoelasticity [40]:
Maxwell (a) model and Kelvin (b) model
Elements connected in series represent a Maxwell model, which is commonly used for interpretation of stress relaxation (Figure 7a). The total strain in this system is a sum of the spring and dashpot strain [40]:
dashpot spring
(11)
An overall system stress is equal to each of the components:
