Evidence of sticky contacts between the wormlike micelles (WLM) in

Subscriber access provided by Nottingham Trent University

Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers

Evidence of sticky contacts between the wormlike micelles (WLM) in viscoelastic surfactant solutions Heinz Hoffmann, and Herbert Thurn Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b02120 • Publication Date (Web): 23 Aug 2019 Downloaded from pubs.acs.org on August 23, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Evidence of sticky contacts between the wormlike micelles (WLM) in viscoelastic surfactant solutions Herbert Thurn* and Heinz Hoffmann** *

ITS, University of Bayreuth, Universitätsstrasse 30, D-95440 Bayreuth, Germany

**BZKG,

University of Bayreuth, Universitätsstrasse 30, D-95440 Bayreuth, Germany

** To whom correspondence should be addressed. E-mail: Heinz. [email protected], [email protected]

Abstract Many cationic surfactants form highly viscoelastic solutions at concentrations of only a few percent. These solutions contain wormlike micelles which can be several micro meter long. The structural relaxation time s in these solutions can be as long as many seconds and the zero-shear viscosity can be in the range of 106 Pas. Electric birefringence measurements (EB) on such solutions had shown four different relaxation times with increasing concentration. The two shortest ones 1 and 2 in the sec region were due to the alignment of small rod-like or ring like micelles. The third one 3 was observed in the viscoelastic region in the msec region and finally a fourth one 4 that was the same as the structural relaxation time s. In this paper it is shown that the 3 process is due to the formation and opening of contacts between the WLM. The reason for the contacts is the attraction between the WLM micelles which is due to the hydrophobic surfaces of the micelles. The contacts crosslink the WLM and thus form a three-dimensional network. This network is the reason for the high viscosity and viscoelastic properties of the samples. If the attraction between the WLM is reduced by adding glycerol to the solution the viscosity of the solution breaks down. The same happens if the surface of the WLM is made hydrophilic by the addition of small amphiphilic molecules. The WLM are not destroyed by these procedures.

1 ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 39

Introduction It is known that many surfactants form wormlike micelles (WLM) 1,2,3. Solutions with WLM are usually viscous or even viscoelastic 4,5. Cryo-TEM measurements on such solutions have shown that the WLM in such solutions can be several m long 6,7. It is usually assumed that the viscosity in these solutions is due to entanglement networks of the wormlike micelles. It also has been shown that the viscosity increases with the concentration by power laws having different exponents which can be as high as 10 and as low as 1.3 8. In special cases negative exponents have even been observed 9. Some of the exponents have been explained by different mechanisms. The structural relaxation times in the viscous fluids can be kinetically controlled by the breaking of the micelles 10, or as in polymer networks, by a reptation process 11. In the first case the viscosity is due to a product of a single shear modulus G° and a single relaxation time (° = G° ). Such solutions are called Maxwell fluids. In entanglement-controlled fluids the structural relaxation process follows a stretched exponential decay. Viscous surfactant solutions have been investigated by SANS and SAXS measurements 12, by NMR 13, by Cryo-TEM 6,7, by DLS 14, by rheology 15 and by electric birefringence measurements16. The most detailed information is obtained from Cryo-TEM measurements. But even this method does not explain why the viscosity follows different power exponents. In this article we present old electric birefringence measurements and reinterpret the old results 17. The new interpretation makes it likely that the WLM in viscoelastic solutions form sticky contacts that form and close. The sticky contact crosslink the WLM and form a dynamic three-dimensional network. This network is responsible for the high viscosity of the solutions and their elastic properties. The sticky contacts are caused by attractive forces between the WLM that have hydrophobic surfaces.

Experimental results The system C8F17SO3 . N (C2H5)4 Perfluoro surfactants with weakly binding counter-ions form globular micelles while with strongly binding counter-ions form wormlike micelles (WLM)18. Solutions of the system C8F17SO3N(C2H5)4 have been studied with different methods 19,20. Conductivity measurements have shown that the system has a CMC of about 1 mM and that 90 % of the counter-ions are strongly bound18. The surface of the WLM are therefore completely covered with tetraethylammonium-ions. Even though the WLM are still negatively charged, the WLM have a hydrophobic surface. This fact will become important when the interaction of the WLM with each other will be discussed.


Page 3 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Detailed rheological and electric birefringence measurements are given in ref.20 The viscosity of the solutions passes from the water viscosity at 1 mM with increasing concentration above a maximum at 60 mM (see fig. 1). Such a behaviour has been observed for many surfactant systems 21. The maximum is due to a switch of the structural relaxation time (s) of the solutions from a reptation controlled mechanism of the WLM to a breaking controlled mechanism of a three-dimensional network. To the left of the viscosity maximum long overlapping WLM exist while on the other side of max the WLM have merged to a three-dimensional network in which endcaps of WLM no longer exist. A diffusion or reptation mechanism is therefore no longer possible.

Fig. 1. Zero/shear viscosity of aqueous solutions of C8F17SO3N(C2H5)4 as a function of concentration at T=25°C.

Electric birefringence measurements on the solution show different relaxation times (fig. 2). Two relaxation times with sec occur only in the dilute low viscous solutions where the WLM do not overlap 19.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 39

Fig. 2. Relaxation time constants τ1 through τ4 vs. concentration for aqueous solutions of C8F17SO3N(C2H5)4 at 25°C . They are due to the alignment of the micelles in the electric field. During the 1-process a part of the short wormlike micelles are aligned parallel to the electric field and during the 2 process a part of the micelles are aligned perpendicular to the electric field. The 2-process can only be observed within a narrow concentration region. Surprisingly, the 1-process can also be observed in the viscoelastic region in which a threedimensional network of the WLM exists. This effect is very likely due to the presence of small ring-like micelles which have been produced from rod-like micelles and which are in equilibrium with ring-like micelles (see in fig. 3). The longest relaxation time (4) is the structural relaxation time s which can also be detected by rheological measurements and is in the range of many seconds. In addition, a relaxation time 3 in the msec region is observed which has been explained differently by different groups and for different systems. The 3 – effect begins at the concentration of 12 mM, that is at the same concentration at which the first contacts are shown in ref. 19 of this manuscript. This is very remarkable because the 3-effect was measured in the bulk phase while the contacts were measured by Cryo-TEM in a thin layer which was only about 50 nm thick. This is a clear sign that the 3 – effect has to do something with the contacts as will be shown later. The peak of 3 corresponds to the concentration at which the viscosity has an inflection point. It is likely that this is no coincidence because the contacts make a contribution to the viscosity. A quantitative relation between the value of 3 and the value of the viscosity for different concentrations is very complex, and at present beyond our means to calculate it.


Page 5 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Fig. 3. Cryo-TEM micrographs of the structures formed by C8F17SO3N(C2H5)4 at 30mM. Bar=100nm. The blue circles show the crosswise contacts, discussed in this paper. The angle of contact is around 90°, which is assumed to be due to electrostatic repulsion. The red circles show branches with three arms and angles around 120° between the branches. Also, the angles are assumed to be due to the electrostatic repulsion (Bar 100nm). Figure 3 shows a section of a CryoTEM micrograph published in ref.19 (Reprinted in part with permission from Ionita-Abutbul I.;

Abezgauz L.; Danino D.; Hoffmann H. Rings and loops in perflurosurfactants viscoelastic solutions. Colloids and Surfaces A: Physicochem. Eng. Aspects 2015, 483, 150-154, Copyright © 2015 Elsevier).

A Cryo-TEM micrograph of a 30 mM solution is shown in fig. (3). The micrograph shows many long entangled WLM with a few small circular micelles trapped in the network. Of particular interest is the fact that some WLM are crossing each other. The angle between two crossing micelles is usually about 90 °. In the two-dimensional micrograph it cannot be decided whether the two micelles are directly in contact or whether the micelles are separated from each other and there is water in between them. This point will be of relevance in the theoretical discussion. Of particular interest will be the angle between two micelles at the crossing point.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 39

The system Cetyltrimethylammonium-Tosylate (CTAT) The phase diagram and the properties of the solution of this system has been studied by J. F. A. Solera et. al 22. The CMC is about 0.1 %. In the range between 0.1 % and 0.4 % globular micelles are observed with a dissociation degree of 25 % are observed. At higher concentration the system forms viscous solutions with WLM. Electric birefringence on this system were made by the group of E. Mendes et. al. 23. They observed again four different relaxation times as had been observed for the system C8F17SO3N(C2H5)4. The 1 process in the dilute region was in the sec time region and increased linearly with the concentration. It was concluded that the process was due to rod-like micelles whose length increases with the concentration. Within a small concentration region, the 2 effect was observed which had an opposite sign as the 1 effect and was a factor 10 – 20 longer as 1. From the overlap concentration the 3 and the 4 effect were observed. The 4 effect had the same time constant as the structural relaxation time s of the viscous system. The authors present a model for the 2 process that goes back to M. Cates and assumes that the 2 process is due to disc-like clusters of WLM 24. This model could not be confirmed by Cryo-TEM measurements on other systems that show the 2 process. Unfortunately, the authors did not discuss a model for the 3 process.

The system C16C8N(CH3)2Br The phase diagram and the properties of its solution have been published in ref. 25. The dissociation degree of the WLM in the dilute regime is 12 %. The isotropic phase with WLM is followed by a L-phase. This is unusual because a phase with WLM is usually followed by a hexagonal Lc-phase. The electric birefringence measurements are given in ref. 26 Both transient electric birefringence (TEB) and dynamic electric birefringence (DEB) measurements were carried out. The results agree with each other. The four relaxation times are observed again as for the three other presented systems (fig. 4).


Page 7 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

1.

Fig. 4. Relaxation time constants τ1 through τ4 vs. concentration for aqueous solutions of C16C8DMABr at T=25°C .(Reprinted with permission from Hoffmann H.; Krämer U.; Thurn H. Anomalous behaviour of micellar solutions in electric birefringence measurements. J. Phys. Chem. 1990, 94, 2027-2033, Copyright © 1990, American Chemical Society)

Surprisingly however was the sign of the birefringence that was opposite to the sign of the mentioned first two systems. This is likely a result of the fact that the WLM of the first two systems had an intrinsic birefringence while the WLM of C16C8N(CH3)2Br were optically isotropic and showed in the aligned state a form birefringence. The anomalous effect with the negative birefringence could only be detected in the small concentration region between 20 mM and 35 mM. The 3 process could be detected following the concentration region after the 2 region. With increasing salt concentration, the different regions of the relaxation times are shifted to smaller concentration.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 39

The system Tetramethylammonium Perfluornonancarboxylat (C8F17.CO2N(CH3)4) Measurements on the system (C8F17.CO2N(CH3)4) revealed also four relaxation processes 27. A signal that contains the first three processes is shown in fig. (5).

Fig. 5. Electric birefringence experiment with three different effects of opposite sign (70mM) C8F17CO2N(CH3)4 , E=4.6x105V/m and T=25°C). The quadratic mode was used for the detections of the signal. The three processes can only be observed within a small concentration region around 70 mM. Most of the N(CH3)4 counter-ions are bound tightly on the surface of the micelles and makes the surface of the micelles hydrophobic 28. It is interesting to note that the monomethyl and dimethyl ammonium counter-ions are even stronger bound as the tetramethyl ammonium ions. This is probably a consequence of the fact that hydrogen bonding is involved in the binding of methyl and dimethyl ammonium ions. It is likely that these counter-ions form vesicles with the perfluoro-ions. The ammonium-ions are strongly dissociated with  = 0.48 and form globular micelles 28.

Further evidence for the formation of contacts between the WLM a) by the influence of the refractive index of the solvent The discussed electric birefringence results are strong evidence that the WLM form mutual contacts what results in a three-dimensional network of the WLM. The results have also shown that the network is not a permanent one but a dynamic one. We thus have an equilibrium between open and closed contacts. In a general sense such an equilibrium is a result between attractive and repulsive forces. The attractive forces can be due to hydrophobic forces and Van der Waals forces. The attractive Van der Waals forces depend on the difference between the refractive index of the WLM and the refractive index of the solvent. This difference can be adjusted by the addition of cosolvents like glycerol 29. Glycerol is a solvent which has little effect on the micelles. 8 ACS Paragon Plus Environment

Page 9 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Previous measurements have shown that the CMC’s of surfactants are little influence by the addition of glycerol to water while the refractive index difference between the WLM and the solvent disappears completely when 60 % of glycerol is added to water. This decrease should have a strong influence on the attraction between the WLM and should lower the viscosity of the solutions. Recent measurements have indeed shown that the viscosity of viscoelastic solutions decreases several orders of magnitude when increasing amounts of glycerol are added to water while the WLM do not disappear 30. b) by the influence of the surface of the WLM The electric birefringence measurements have shown that contacts between the WLM are formed when the WLM have a hydrophobic surface. The contacts should therefore disappear and the viscosity of the solutions should decrease when the surface of the micelles is made hydrophilic. In principle this could be done by substituting the methyl groups on the surfactants by CH2OH groups. Unfortunately, surfactant with such head groups are commercially not available and measurements on such systems could not be made. However, it is also possible to change the hydrophobic surface of the WLM by adding small amphiphilic molecules like isopropanol or 1.3 dihydroxybutane to the micellar solution 31. These small molecules bind to the hydrophobic surface of the micelles and make it hydrophilic. As a consequence, the WLM can no longer bind to each other and the viscosity has to decrease. Recent experiments have confirmed these expectations 31.

Theoretical interpretation The experimental results show that the amplitude of the 𝜏3-effect begins when the WLM have reached a length from which they overlap and the viscosity begins to rise steeply. We can assume therefore that around this critical length of the WLM the contacts begin to form. For simplicity of the model we assume further that for higher concentration an equilibrium exists in solution between free WLM and WLM with a contact in between them. Such a situation is shown in fig. (6). In a more abstract sense, very long WLM may consist of a number of statistical independent parts with a persistence length 𝐿𝑝 48,49.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 39

For such an equilibrium the change of the free WLM concentration [M1] is given by equation (1) 𝑑[M1] 𝑑𝑡

= ― k12[M1]2 +2k21[M2]

(1)

where k21 is the decay rate of contact concentration [M2] . For such a simple system the reciprocal relaxation time 𝜏 will be given by equation (2): Fig. 6. Schematic picture of the kinetics of the micellar contacts.

1

τ = k21 +2k12[M10]

(2)

where [M10] denotes the equilibrium value of [M1] which may depend on the surfactant concentration in a complicated way. On the other hand, we know from the electric birefringence measurements, that 𝜏3 does not vary much with the surfactant concentration, so that we may assume that 𝜏3 is mainly dominated by the decay rate k21. Therefore, we may write: τ3 =

1 k21

(3)

Furthermore, we assume that the probability of a decay can be described by a Boltzmann factor 𝑒 has:

―

𝐸𝑎𝑐𝑡 𝑘𝑇

, where 𝐸𝑎𝑐𝑡 is the activation energy to open the contact, so one

k21 = k021e

―

Eact kT

(4)

In the case of εact = 0, no energy is necessary to open the contact and the WLM are allowed to rotate approximately free in the same way as micelles from concentration regime [M1]. So, one has:

τ1 =

1

(5)

k021

From (3) with (4) and (5) one finally gets: 𝜏3 = 𝜏1 𝑒


(6)

It also interesting to note that the sign of birefringence of τ1 and τ3 is always the same in our experiments, so we may assume the same type of micellar structure as an origin for 10 ACS Paragon Plus Environment

Page 11 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

the measured birefringence. In the case of C8F17SO3N(C2H5)4 the sign of the birefringence is negative as is shown in figure 2. Equation (6) is used to interpret our birefringence data. It should be mentioned that similar considerations were done by H. Eyring who made a theory of transport phenomena in dense gases and liquids 51 where a non-Newtonian behaviour of the viscosity can be calculated due to an activated state.

Table 1. 𝑬𝒂𝒄𝒕/𝒌𝑻 calculated from birefringence data by equation (6) for 𝐂𝟖𝐅𝟏𝟕𝐒𝐎𝟑 𝐍(𝐂𝟐𝐇𝟓)𝟒 c in 15 mM 𝜏1 𝑖𝑛 𝜇𝑠 2.0

20

25

30

35

40

45

50

60

1.5

1.2

1.1

1

0.95

0.9

0.85

0.8

𝜏3 𝑖𝑛 𝑚𝑠 1.6

3

3

4

4

3

2.2

2

1.5

7.6

7.82

8.2

8.29

8.06

7.8

7.76

7.5

𝐸𝑎𝑐𝑡

6.70

𝑘𝑇

For the System C16C8DMABr a similar behaviour can be measured. In the case of C16C8DMABr (see fig. 4) both τ1 and τ3 have positive birefringence. The calculated values of 𝐸𝑎𝑐𝑡/𝑘𝑇 are shown in the next table.

Table 2. 𝑬𝒂𝒄𝒕/𝒌𝑻 calculated from birefringence data by equation (6) for 𝐂𝟏𝟔𝐂𝟖𝐃𝐌𝐀𝐁𝐫 c in mM τ1 in μs τ3 in ms

40 0.9 1

42 1.5 2

45 1.3 2.5

60 1.5 3.7

𝐸𝑎𝑐𝑡

7.0

7.25

7.56

7.8

𝑘𝑇 In order to compare the calculated values of 𝐸𝑎𝑐𝑡/𝑘𝑇 of table 1 and 2 with values from a theoretical model, this model is described in the next section.

Theoretical model


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 39

To keep the things simple, we have to make some assumptions, that help us formulate the mathematics behind. First, the wormlike micelle is approximated by a rod (see fig. 7). The length of the rod is in the order of the persistence length 𝐿𝑝 of the worm like micelle. This is not a too servere restriction as we can see from the Cryo-TEM pictures, i.e. a very long WLM is treated as consisting of statistically independent micelles with persistence length 𝐿𝑝48.49.

Fig. 7. Approximation of the crosswise contact by the crosswise contact of two rods Second, the rods are segmented in 𝑁1 +1 and 𝑁2 +1 parts, where each segment 𝑙𝑠 interacts with the other segment of the other rod while the unit vectors of orientation u1 of rod 1 and u2 of rod 2 is taken full into account.

Fig. 8. Definition of the full vector identity 12 ACS Paragon Plus Environment

Page 13 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

From figure 8 the full vector identity is given by: 𝑟𝑖𝑗 = 𝑟𝑖𝑗(𝑅,𝑢1,𝑢2) =

with ―

𝑁1 2

≤𝑖≤

𝑁1 2

and ―

𝑁2 2

≤𝑗≤

𝑁2 2

(7)

𝑅 + 𝑖 𝑙𝑠 𝑢1 + 𝑗 𝑙𝑠 𝑢2

, while 𝑅 is the vector between the centres of

the rods. For symmetry reasons one has 𝑟00 = 𝑅 ; hence the centre of the rod is described by the index 0. To keep the things simple, we use 𝑁1 = 𝑁2, which means, that the rods have the same length. Third, for each segment it is assumed, that there exists a point to point interaction which can be written as a sum of an electrostatic 𝑊𝑒(𝑟𝑖𝑗) and the van der Waals energy 𝑊𝑣𝑑𝑊 (𝑟𝑖𝑗) while the hydrophobic interaction 𝑊𝐻 only acts at distance 𝑅 , which can be described by a hydrophobic contact energy 𝐸𝐶 at the center of mass of the rod-like particles. Therefore, the total energy W𝑡𝑜𝑡 depends on the vectors 𝑅,𝑢1,𝑢2 . As we use the full vector identity, each three-dimensional configuration of the rods within its physical limitations can be calculated. 𝑁 2

W𝑡𝑜𝑡(𝑅,𝑢 1,𝑢2) = 𝑊𝐻(𝑅) +

𝑁 2

∑ ∑𝑊(

𝑒 𝑟𝑖𝑗(𝑅,𝑢 1,𝑢2))

+ 𝑊𝑣𝑑𝑊(𝑟𝑖𝑗(𝑅,𝑢 1,𝑢2))

𝑁 𝑁 𝑖=― 𝑗=― 2 2

with |𝑟𝑖𝑗| > 𝑅𝑚𝑖𝑛 W𝑡𝑜𝑡(𝑅,𝑢 1,𝑢2) = ∞ for

and |𝑟𝑖𝑗| ≤ 𝑅𝑚𝑖𝑛 (Hard Sphere Interaction) (8)

𝑅𝑚𝑖𝑛 is defined in the following sections (see equation (17)). For symmetry reasons, in the case of rods, the following identities hold:

W𝑡𝑜𝑡𝑎𝑙(𝑅,𝑢 1,𝑢2) = W𝑡𝑜𝑡𝑎𝑙(𝑅, ― 𝑢 1,𝑢2) = W𝑡𝑜𝑡𝑎𝑙(𝑅,𝑢 1, ― 𝑢2) = W𝑡𝑜𝑡𝑎𝑙(𝑅, ― 𝑢 1, ― 𝑢2) (9) The main idea behind this ansatz is, that we describe each particle shape by a set of vectors and put point to point interactions on its shape. For the interactions only depend on the scalar 𝑟𝑖𝑗(𝑅,𝑢 1,𝑢2) = 𝑟𝑖𝑗 ∙ 𝑟𝑖𝑗 , where ∙ denotes the scalar product of the vector 𝑟𝑖𝑗, the total energy W𝑡𝑜𝑡 can easily be calculated by adding up all contributions. With respect to the numerical analysis, W𝑡𝑜𝑡 depends on the number 𝑁 of summands. In order to have a criterion how large 𝑁 should be to reach a certain accuracy, the following inequality is used:

|

𝑊𝑡𝑜𝑡(𝑁 + 2) ― 𝑊𝑡𝑜𝑡(𝑁) 𝑊𝑡𝑜𝑡(𝑁 + 2)

| < 10

―3


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 39

In the cases discussed here, 𝑁 = 17 is sufficient to reach this accuracy. From the physical point of view this results in a particle size of 100 – 120 nm (N*𝑙𝑠 , see equ. (14)), which is consistent with Cyro-TEM micrograph of fig. 3.


Page 15 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

The electrostatic interaction For the electrostatic interaction energy will be assumed, that a cylindrical segment of length 𝑙𝑠 and radius 𝑅𝐶 may be described by a virtual spherical particle of Radius 𝑅𝑆. The screening of the effective charge 𝑍𝑒𝑓𝑓 of the particles is described by a surrounding 1 ionic cloud, which is characterized by the Debye length 𝜅 (see fig.9). The counterion distribution is considered to be quasi continuous.

Fig. 9. Each cylindrical segment is approximated by a virtual sphere. The dotted lines symbolize the range of the Debye length.

For the interaction of two spherical, screened particles with effective charge 𝑍𝑒𝑓𝑓 one has: 2𝜅𝑅

𝑊𝑒(𝑟𝑖𝑗) = where and

―𝜅𝑟

2 1 𝑍𝑒𝑓𝑓 𝑒 𝑠 𝑒 𝑖𝑗 4𝜋𝜀𝜀0 (1 + 𝜅𝑅𝑠)2 𝑟𝑖𝑗

(9)

𝜀0 is the dielectric constant 𝜀 is the dielectric number of the medium.

The effective charge 𝑍𝑒𝑓𝑓 and the Debye length

1

𝜅 of the virtual sphere is given by:

𝑍𝑒𝑓𝑓 = 𝛼 𝑛𝑎𝑔𝑔 𝑒 𝜅= 𝐼=

2 𝑁𝐿𝑒2𝐼 1000 𝜀𝜀0 𝑘𝑇 𝛼 𝑐𝑚𝑐 + 2 (𝑐

(10) (11)

― 𝑐𝑚𝑐)

(12)


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 39

Where 𝛼 is the degree of dissociation, 𝑛𝑎𝑔𝑔 is the aggregation number of the segment, 𝑒 is the elementary charge, 𝐼 is the ionic strength, 𝑐𝑚𝑐 is the critical micellar concentration and 𝑐 is the molar concentration of the surfactant. The degree of dissociation is assumed to be between 3% and 10%, which is known from conductivity measurements. For the segment length 𝑙𝑠 and 𝑅𝑆 always the inequalities 32 𝑙𝑠
8 𝑘𝑇 makes no sense for a stable contact. Furthermore, as already stated, 𝐸𝐶 and 𝐿𝐻 cannot be changed within one series of measurements, i.e. there cannot be different values of 𝐸𝐶 and different values of 𝐿𝐻 in one series of measurement, when the surfactant concentration is increased. To get better insight, what might happen, if two charged rods like particles approach each other in the relative energy minimum of a crosswise orientation, figure 13 shows the influence of an increasing hydrophobic energy 𝐸𝐶 on the total energy.


Page 23 of 39

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

(b)

(c)

(d)

Page 24 of 39

Fig. 13 (a) A simple description of the geometry. One rod is fixed, while the other is perpendicular oriented and moves around in the x – y plane with different center to center distances 𝜅𝑅. (b) Energy surface, with no hydrophobic area 𝐸𝐶 = 0𝑘𝑇. (c) Energy surface, with 𝐸𝐶 = ―10𝑘𝑇. (d) Energy surface, with 𝐸𝐶 = ―14𝑘𝑇. The other parameters are the same as in the previous figures. The red bars symbolize the two rods-

As can be seen by figure 13(b) the charged rods cannot get into contact without additional hydrophobic attraction at the centers. When the hydrophobic energy is increased (figure 13(c)) the rod-like particle may have contact, but the contact is not stable, because of 𝐸90 > 𝐸𝑎𝑐𝑡. Figure 13(d) shows the same situation as in figure 13(c) but with a stable contact because of 𝐸90 < 𝐸𝑎𝑐𝑡 . In order to explain an activation energy of about 8 𝑘𝑇 which we get from the electric birefringence measurements, we use plausible values as in figure 11 and 12, and one should keep in mind, that the only varied parameter is the degree of dissociation, while the other parameters are held constant. A detailed discussion of the multidimensional energy surface is beyond the scope of this paper. 24 ACS Paragon Plus Environment

Page 25 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Results and discussion The system 𝐂𝟖𝐅𝟏𝟕𝐒𝐎𝟑𝐍(𝐂𝟐𝐇𝟓)𝟒.

Fig. 14 Activation energy and energy barrier in units of kT versus the surfactant concentration for C8F17 SO3N(C2H5)4 with the influence of the errors of 𝜏1 and 𝜏3, where the relative error is assumed to be

smaller or equal than 10% (see appendix B). The calculated value at around 15 mM may be due to a measuring error as 𝜏3 is hard to measure in that range. The error limits show a smooth transition region (shadowed area) from I to II that lies between 11 and 19 mM and not a sharp border. This corresponds to the fact, that there are only few crossing contacts seen in the Cryo-TEM pictures in that range.

I)

𝜏3 not measurable

II) 𝜏3 ≤ 𝜏3𝑚𝑎𝑥

III) 𝜏3 ≥ 𝜏3𝑚𝑎𝑥

It should be noted that the lower limit of the transition range corresponds with the region, where the viscosity starts to rise (see fig.1), while the transition from II to III seems to correspond with the inflection point of the viscosity. The solid lines are theoretical calculations. The little squares are experimental values, which are calculated by equation (7). The little circles denote calculated values from the experimental values. In region I the activation energy is always smaller than the energy barrier 𝐸𝑎𝑐𝑡 < 𝐸90, hence there are no stable contacts and 𝜏3 cannot be measured. This also means, that the hydrophobic and the van der Waals energy are always smaller than the electrostatic repulsion, so that the total energy is always positive. In the transition range from I to II, at around 15 mM we have a relatively large deviation of 10% from the experimental value

𝐸𝑎𝑐𝑡

𝐸𝑎𝑐𝑡

𝑘𝑇

𝑘𝑇

( )𝑒𝑥𝑝 = 6.7 with respect to the theoretical one of ( )𝑡ℎ𝑒𝑜𝑟. = 7.25, which may

indicate, that the 𝜏3-effect is difficult to measure in this transition range as the signal is rather weak when there are only a few contacts. 25 ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 39

In region II, from 20 mM to 30mM, where 𝜏3 ≤ 𝜏3𝑚𝑎𝑥 all the theoretical calculation can be done with one set of parameters; i.e. the dependence on the concentration is simply given by the ionic strength (see equ. 12) and we have a very good agreement between experimental and theoretical data. The parameter set for region II is given by:

{𝐴

𝐻

}

= 0.82 𝑘𝑇, 𝛼 = 3.40%, 𝑅𝑐 = 1.4 𝑛𝑚, 𝑐𝑚𝑐 = 0.9 𝑚𝑀𝑜𝑙, 𝐸𝐶 = ―15.6 𝑘𝑇, 𝐿𝐻 = 3.67 𝑛𝑚, 𝑇 = 298𝐾, 𝜀𝑟 = 78.4

An example is given with the following picture:

Fig. 15 Theoretical calculation for C8F17SO3N(C2H5)4 at 25 mM. An enlarged section is given in the right upper corner. The contribution of the van der Waals energy to 𝐸𝑎𝑐𝑡 is around 4 𝑘𝑇.

At very small distances the total energy is always negative; so, we have stable contacts. The difference between the energy maximum WLMs and the activation energy


𝐸90 𝑘𝑇

with a perpendicular alignment of the

is around -2kT (see fig. 15) and increases with

increasing concentration as can be seen in figure 14. Furthermore, it may be interesting to note, that for all calculations done in this paper, the range of the hydrophobic energy 26 ACS Paragon Plus Environment

Page 27 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

is always smaller than the Debye length. Hence the dominating energy is the electrostatic repulsion. But, for an alignment around 90°, this energy may be compensated by the hydrophobic and the van der Waals energies to form contacts. This is in agreement with the cryo-TEM pictures. In region III, where 𝜏3 ≥ 𝜏3𝑚𝑎𝑥 and 𝜏4 ≠ 0, with the exception of 𝛼 all the theoretical calculation can be done with the same set of parameters as in region II.

{𝐴

𝐻

= 0.82 𝑘𝑇, 𝑅𝑐 = 1.4 𝑛𝑚, 𝑐𝑚𝑐 = 0.9 𝑚𝑀𝑜𝑙, 𝐸𝐶 = 15.6 𝑘𝑇, 𝐿𝐻 = 3.67𝑛𝑚, 𝑇 = 298𝐾, 𝜀𝑟 = 78.4

}

This means, that the hydrophobic and the van der Waals energies have the same influence as in region II and depend only on the nature of the surfactant. In order to achieve a good agreement of experimental and theoretical data, 𝛼 has to be described with slight dependence from the concentration ( 𝛼 = 0.2618𝑐 + 0.0257 ) with c in Mol/liter) additionally to the concentration dependence of the ionic strength. This may be due to the increasing entanglement of the wormlike micelles, which form a threedimensional network. So, our simple assumption of pairwise interacting WLMs is no longer correct and the influence of the neighbouring micelles may be described with an additional concentration dependence of 𝛼. Furthermore, it is interesting to note, that in region III the energy barrier is approximately constant with a value around 5 𝑘𝑇, while the activation energy slightly decreases with concentration.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 39

The system 𝐂𝟏𝟔𝐂𝟖𝐃𝐌𝐀𝐁𝐫.

Fig. 16 Activation energy and energy barrier in units of kT versus the surfactant concentration for C16C8DMABr. The theoretical calculations are omitted in region III as we have only one measuring point in this region. The magnitude of the error is similar as shown in fig. 14 and not plotted.

The parameter set for region II is given by:

{𝐴

𝐻

}

= 0.81 𝑘𝑇, 𝛼 = 3.95%, 𝑅𝑐 = 1.8 𝑛𝑚, 𝑐𝑚𝑐 = 1.3 𝑚𝑀𝑜𝑙, 𝐸𝐶 = ―14 𝑘𝑇, 𝐿𝐻 = 4.8 𝑛𝑚, 𝑇 = 298𝐾, 𝜀𝑟 = 78.4

The concentration dependence of the system C16C8DMABr in the region II (see fig. 16) can be described by one fixed set of parameters similar to the system C8F17SO3N(C2H5)4 and is simply given by the ionic strength according equation (12), which enters the electrostatic repulsion due to the Debye length. In region III, from the measured value of 𝐸𝑎𝑐𝑡 𝑘𝑇

= 7.8, at c=60mM, the energy barrier

𝐸90 𝑘𝑇

= 3.327 is calculated for an 𝛼 = 4.480%.

The other parameters are the same as in region II. The result is a higher 𝛼 value than in region II. So, the tendency is the same as for system C8F17SO3N(C2H5)4 and hence the increase of 𝛼 may be interpreted by the forming of a three-dimensional micellar network.


Page 29 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Discussion of the results Although the chemical structure of the systems C8F17SO3N(C2H5)4 and C16C8DMABr is rather different, they show a similar behaviour in electric birefringence and rheological measurements. In the region, where the 𝜏3- effect can be measured and where 𝜏3 ≤ 𝜏3𝑚𝑎𝑥 the concentration dependence can be described only by the ionic strength of the systems, while the other parameters are constant. The results are shown in the following table.

Table 3. Parameters calculated from birefringence data for 𝐂𝟏𝟔𝐂𝟖𝐃𝐌𝐀𝐁𝐫 and 𝐂𝟖𝐅𝟏𝟕𝐒𝐎𝟑𝐍(𝐂𝟐𝐇𝟓)𝟒 C8F17SO3N(C2H5)4

C16C8DMABr

0.82 kT 3.67 nm -15.6 kT 0.0340

0.81 kT 4.8 nm -14 kT 0.0395

𝐴𝐻 𝐿𝐻 𝐸𝐶 𝛼

For the hydrocarbon surfactant, the Hamaker Constant 𝐴𝐻 is in in good agreement with other hydrocarbon-systems 50 both theoretically and experimentally. For the fluorocarbon surfactant, 𝐴𝐻 is nearly the same. The other parameters like the hydrophobic range 𝐿𝐻, the energy of contact 𝐸𝐶 and the degree dissociation 𝛼 are also in the same order of magnitude in both systems. In the case of the fluorocarbon surfactant 𝐿𝐻 is comparable with the short-range decay in ref. 39 Furthermore, as already shown in fig. 14, due to the measuring error of quantities 𝜏1 and 𝜏3, the calculated parameters in table 3 vary by about 3%; e.g. a variation of 𝜏1 and 𝜏3 up to 10% results in variation of 𝛼 by 3%. If one further assumes, that the following relation between the surface tension 𝛾, the area of contact 𝐴𝑟 and the contact energy 𝐸𝐶 holds: 𝐸𝐶 = 2 𝛾𝐴𝑟

one gets

𝐴𝑟 = 𝐸𝐶/2𝛾

𝑚𝐽

So, as 𝛾 is in the range of 40 ― 50 𝑚2 for hydrocarbons with water 50 one gets for the range of the contact area 0.57 – 0.72 𝑛𝑚2, while the area of the virtual sphere rep. the 𝐴𝑟

cylindrical segment is 91.61 𝑛𝑚2, so always the relation 4𝜋𝑅2 ≪ 1 holds. 𝑠

The aim of this manuscript is to demonstrate that the WLM in viscoelastic surfactant solutions form dynamic contacts with each other that constantly form and open again. 29 ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 39

The experimental evidence for this assumption was the relaxation process 3 that is observed in electric birefringence measurements. It was assumed that this process is due to the shift of the equilibrium between the open and closed contact when an electric field is applied to the system. The situation is thus very similar as for the appearance for the 4 process in electric birefringence measurements. During this process entangled wormlike micelles align in the electric field. This process is several orders of magnitude slower as 3. We would like to mention that as far as we know this is the first time in which contacts between wormlike micelles have been proposed in viscoelastic surfactant solutions. The reason for this is probably the general assumption that contacts between ionically charged WLM micelles cannot exist. In the next step it was assumed that the contacts can only contribute to the total electric birefringence when the contact opens. It is only then when the new free WLM can orient in the electric fields. In order to open the contacts, the thermal energy has to break the 𝐸𝑎𝑐𝑡 𝑘𝑇

contact energy. This allows us to write 𝜏3 = 𝜏1 𝑒 where 1 is the rotation time for the free WLM. Furthermore, this equation allows us to calculate an activation energy purely from the two relaxation times 3 and 1. The result was 𝐸𝑎𝑐𝑡 ≈ 7… 8 𝑘𝑇 and can be compared with the contact energy 𝐸𝐶 that can be calculated from the hydrophobic energy between hydrocarbon and water 44 what results in 𝐸𝐶 (-15.6 kT …-14 kT) depending on the hydrocarbon (s. Tab. 3). So 𝐸𝑎𝑐𝑡 is about the half of 𝐸𝐶. This is due to the influence of the electrostatic repulsion. Additionally, the associated range 𝐿𝐻 of the hydrophobic attraction energy is in good agreement known for similar systems 39,47. Furthermore, the calculated Hamaker constant 𝐴𝐻is consistent with values that can be calculated for such systems by the Lifshitz theory. Considering that the surface of the WLM contains besides the methyl groups of the head groups also some polar group, the two thus obtained values are in satisfactory agreement. This can be taken as evidence that the interpretation of the results is consistent. In the theoretical part, the different energies that are involved in the contact energy as the electrostatic energy, the Van der Waals energy and the hydrophobic energy were calculated as a function of the distance between the WLM and their mutual orientation. These calculations showed that the contribution of the Van der Waals energy to the total energy with the exception of very small distances can be neglected. On large separation between the WLM the repulsive energy is larger than the attractive energy that is due to the hydrophobic energy. On short scale (𝜅𝑟 ≤ 1) the hydrophobic energy together with a 90° degree orientation in a relative energy minimum overcomes the energy barrier formed with the electrostatic energy and leads to an on approach of the two WLM to form a contact. On contact the WLM are stable oriented perpendicular to each other. This is 30 ACS Paragon Plus Environment

Page 31 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

obvious in fig. 3. The angle of 90° of crosslinks in Cryo-TEM pictures as in the system of Gemini surfactants 45 can therefore be taken as evidence of contacts between their WLM. The crosslinking of the WLM leads to a three-dimensional network of the micelles and hence to an increaser of the zero-shear viscosity ° in comparison to systems in which the WLM are not crosslinked. It thus becomes clear why the viscosity for some surfactants increases so strongly with the concentration with a power law exponent of 10 while in systems that also form WLM that do not crosslink the viscosity increases only modestly with the concentration. In summary we can therefore conclude that viscoelastic surfactant solutions contain WLM which are cross linked and the angle at the contacts between the WLM is 90 ° due to electrostatic repulsion. The origin for the crosslinks is of course the hydrophobic surface of the WLM. Surfactants that can form WLM with hydrophobic surfaces are CTAB, alkylpyridinium surfactants and alkyldimethylaminoxids. It is likely that solutions from these micelles can be very viscous while surfactants with hydrophilic head groups form micelles with hydrophilic surfaces and therefore form low viscous solutions even when they form WLM. Typical systems that belong to this type are anionic surfactants with the head groups CO2- or SO3 -. Finally, it should be mentioned that the 3-process that can be observed in electric birefringence measurements on which all the observed conclusions were based, can also be observed by oscillating rheological measurements. However, since 3 is in the range of msec, this is only possible with an instrument which is capable to measure up to an angular frequency of  = 1000𝑠 ―1 , see ref.46,52,53 It may be interesting to have high frequency rheological measurements on such systems.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 39

Conclusions Electric birefringence measurements in viscoelastic surfactant solutions show four relaxation times. The two shortest relaxation times 1 and 2 in the sec range are due to the alignment of the wormlike micelles in the electric field. The longest one 4 in the range of seconds is the structural relaxation time of the viscosity. It is shown for the first time that the relaxation time 3 in the range of msec is due to the formation of contacts between the WLM even when the WLM are ionically charged. The contacts open and close within msec. The contacts consist of two WLM that are aligned perpendicular to each other and are clearly visible in Cryo-TEM micrographs. They may be explained by the attraction between the WLM on a nanoscale that have hydrophobic surfaces which are a result of methyl- and ethyl-groups on the head groups of the surfactants. With increasing surfactant concentration, the contacts crosslink the WLM to a three-dimensional network that is responsible for the high viscosity of the solutions. An activation energy 𝐸𝑎𝑐𝑡 is calculated from the relaxation time 3 with the help of the 𝐸𝑎𝑐𝑡

relaxation time 1 by the equation 𝜏3 = 𝜏1 𝑒 𝑘𝑇 . The activation energy can be thought as a combination of the hydrophobic contact energy, the van der Waals energy and the orientation dependent repulsive electrostatic energy between the WLM. The contacts can be weakened or destroyed by the addition of small amounts of amphiphilic molecules that bind on the hydrophobic surface of the WLM or by adding glycerol to the solution. With the loss of the contacts the viscosity of the solution decreases strongly. It is mentioned that the endcaps of the rod-like micelles may also contain hydrophobic parts and hence may also contribute to different particle structures. So, one may have contacts between the ends of two rod-like micelles and so on to form a WLM or an end may have contact with hydrophobic area of another WLM to form a branch with three arms (see fig. 3) or may have contact with its own WLM to form a loop (see fig. 3) or may have contact with its own other end to form a ring like micelle. All of the described micellar structures are seen in cryo-TEM micrographs and can be best explained by the interactions described in this paper.


Page 33 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Appendix A: As the charges sit on the surface of the micelles, we demand that the area of the cylindrical segment and the virtual sphere are the same; i.e. we have: (A1)

4𝜋 𝑅2𝑆 = 2𝜋𝑅𝐶𝑙𝑠 = 2𝜋𝑅2𝐶 𝑥𝑙𝑠

where 𝑥𝑙𝑠 is a factor, which describes the segment-length in units of 𝑅𝑐 ; so, one has (A2a)

𝑙𝑠 = 𝑥𝑙𝑠𝑅𝐶 resp.

𝑥𝑙𝑠

𝑅𝑆 =

(A2b)

𝑅𝐶

2

Additional we demand, that the mass 𝑚 in the sphere is the same as in the cylindrical segment. (A3)

𝑚 = 𝜌𝑆𝑉𝑆 = 𝜌𝐶𝑉𝐶

where 𝜌𝑆 and 𝜌𝐶 are specific densities of the both volumes. Using the equations above, one gets: 2𝑥𝑙𝑠

𝜌𝐶 =

3

𝜌𝑆

And for the assumption 𝜌𝐶 = 𝜌𝑆 one has 𝑥𝑙𝑠 =

9 2

(A4)

and

3

𝑅𝑆 = 2𝑅𝐶

(A5)

To keep the things simple, we always use the equations (A4) and (A5) in our calculations as fixed equations.

Appendix B: In order to give a rough estimation of the error propagation starting from the measured values of 𝜏1 and 𝜏3 the following considerations are made. In our calculations, the logarithm of 𝜏1 and 𝜏3 is used (see equ. 6). Hence a deviation of this value due to relative measuring errors of Δ𝜏1/𝜏1 and Δ𝜏3/𝜏3 is given by: Δln

( )= 𝜏3

Δ𝜏3

𝜏1

τ3

―

Δ𝜏1 τ1

≤

| |+| | Δ𝜏3

Δ𝜏1

τ3

τ1

where Δ𝜏1 and Δ𝜏3 can be both positive or negative and describes the upper and the lower error limit. So, the relative error of the calculated quantity is given by: Δln

( ) / ln ( ) ≤ ( | | + | | )/ln ( ) 𝜏3

𝜏3

Δ𝜏3

Δ𝜏1

𝜏3

𝜏1

𝜏1

τ3

τ1

𝜏1


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 39

From experimental data we know, that in our case ln assume that the relative errors

Δ𝜏1 τ1

and

Δ𝜏3 τ3

( ) ≈ 8. Furthermore, we may 𝜏3 𝜏1

are smaller than 10%. So, we get an upper limit

for the calculated relative error of 𝐸𝑎𝑐𝑡: Δ𝐸𝑎𝑐𝑡 𝐸𝑎𝑐𝑡

= Δln

( ) / ln ( ) ≤ 𝜏3

𝜏3

𝜏1

𝜏1

2 ∗ 10% 8

< 3%

The error limits are shown in fig. 14. Acknowledgment We would like to thank G. Platz for stimulating discussions.


Page 35 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

References: 2.

Kern F.; Lemarechal P.; Candau S. J.; Cates M. E. Rheological properties of semi dilute and concentrated aqueous solutions of cetyltrimethylammonium bromide in the presence of potassium bromide. Langmuir 1992, 8, 437-440.

3.

Chu Z.; Dreiss C. A.; Feng Yujun Smart wormlike micelles. Chem. Soc. Rev., 2013, 42, 7174-7203.

4.

Appell J.; Porte G.; Khatory A.; Kern F.; Candau S. J. Static and dynamic properties of a network of wormlike surfactant micelles (cetylpyridinium chlorate in sodium chlorate brine) J. Phys. II, France 2 1992, 1045-1052.

5.

Magid L. J. The surfactant-polyelectrolyte analogy. J. Phys. Chem. B 1998, 102, 21, 4064-4074.

6.

Brown W.; Johansson K.; Almgren M. Threadlike micelles from cetyltrimethylammonium bromide in aqueous sodium naphthalenesulfonate solutions studied by static and dynamic light scattering. J. Phys. Chem., 1989, 93, 5888-5894.

7.

Clausen T. M.; Vinson P. K.; Minter J. R.; Davis H. T.; Talmon Y.; Miller W. G. Viscoelastic micellar solutions: microscopy and rheology. J. Phys. Chem. 1992, 96, 474-484.

8.

Danino D.; Talmon Y.; Zana R. Alkanediyl-, -bis (dimethylalkylammonium bromide) surfactants (dimeric surfactants). 5. Aggregation and microstructure in aqueous solutions. Langmuir 1995, 11, 5, 1448-1456.

9.

Hoffmann H. Chapter 1 Viscoelastic surfactant solutions. In Herb C. A.; Prudhomme R. K. (eds.), Structure and flow in surfactant solutions. ACS Symposium Series, 1993, 578, 22-27.

10.

Hoffmann H.; Löbl M.; Rehage H.; Wunderlich I. solutions. Tenside Deterg., 1985, 22, 290-298.

11.

Hoffmann H.; Löbl M.; Rehage H. Viscoelastic surfactant solutions: kinetically controlled viscosities. Progress and Trends in Rheology II 1988, supplement to Rheologica Acta, 1988, 26, 246-248.

12.

Lequeux F. Reptation of connected wormlike micelles. Europhys. Lett., 1992,19 (8), 675-681.

13.

Aswal V. K.; Goyal P. S.; Thiyagarajan P. Small-anlge neutron-scattering and viscosity studies of CTAB/NaSal Viscoelastic micellar solutions. J. Phys. Chem. B, 1998, 102, 14, 2469-24473.

14.

Monduzzi M.; Olsson U.; Söderman O. Bicontinuous “micellar” solutions. Langmuir 1993, 9, 11, 2414-2920.

Rheology of surfactant


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 39

15.

Nemoto N.; Kuwahara M. Dynamic light scattering of CTAB/NaSal long threadlike micelles in the semidilute regime: applicability of the dynamic scaling law. Langmuir 1993, 9, 419-423.

16.

Berret J.-F. Transient rheology of wormlike micelles. Langmuir 1997, 13, 8, 2227-2234.

17.

Schorr W.; Hoffmann H. Electric birefringence of micellar solutions. Proceedings of the Intern. School of Physics Enrico Fermi, Physics of amphiphiles: micelles, vesicles and microemulsions 1985

18.

Hoffmann H.; Gräbner D. Electric birefringence anomaly of solutions of ionically charged anisometric particles. Advances in Colloid and Interface Sci. 2015, 216, 20-35.

19.

Hoffmann H.; Tagesson B. The influence of substituted ammonium-ions on the thermodynamics and kinetics of micelles of perfluorated octanesulfonate. Physikalische Chemie Neue Folge 1978, 110, 113-134.

20.

Ionita-Abutbul I.; Abezgauz L.; Danino D.; Hoffmann H. Rings and loops in perflurosurfactants viscoelastic solutions. Colloids and Surfaces A: Physicochem. Eng. Aspects 2015, 483, 150-154.

21.

Angel M.; Hoffmann H.; Krämer U.; Thurn H. The electric birefringence anomaly in colloidal and micellar systems. Ber. Bunsenges. Phys. Chem. 1989, 93, 184-191.

22.

Hoffmann H.; Thunig C.; Schmiedel P.; Munkert U.; Ulbricht W. The rheological behaviour of different viscoelastic surfactant solutions. Tenside Surf. Deterg. 1994, 31, 6, 389-400.

23.

Soltero J. F. A.; Puig J. E.; Manero O.; Schulz P. C. Rheology of cetyltrimethylammonium tosilate-water system. 1. Relation to phase behaviour. Langmuir 1995, 11, 3337-3346.

24.

Narayanan J.; Mendes E.; Manohar C. Anomalous ordering in wormlike micelles under electric field. J. Phys. Chem. 1996, 1000, 18524-18529.

25.

Cates M. E. The anomalous Kerr effect: implications for polyelectrolyte structure. J. Phys. II France, 1992, 2, 1109-1119.

26.

G. Neubauer.; Hoffmann H.; Kalus J.; Schwandner B. The shape of hexadecyloctyldimethylammoniumbromide micelles in aqueous solutions. Chem. Phys. 1986, 110, 247-253.

27.

Hoffmann H.; Krämer U.; Thurn H. Anomalous behaviour of micellar solutions in electric birefringence measurements. J. Phys. Chem. 1990, 94, 2027-2033.

28.

Angel M., Elektrooptisches Verhalten mizellarer Systeme. Bayreuth, 1985

Dissertation


Page 37 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

29.

Hoffmann H.; Platz G.; Rehage H.; Reizlein K.; Ulbricht W. Messungen zum Aggregationsverhalten perfluorierter Alkansäuren. Makromol. Chem. 1981, 182, 451-481.

30.

Song A.; Reizlein K.; Hoffmann H. Swelling of aquous L-phases by matching the refractive index of the bilayers with that of the mixed solvent. Progr. Colloid Polym. Sci. 2008, 134, 111-119.

31.

Hoffmann H.; Abdel-Rahem R. Influence of co-solvent on the rheological behaviour of aqueous viscoelastic surfactant solutions. Colloid Polym. Sci. 2010, 288, 603-612.

32.

Clinckspoor K. J.; Lorenzetti Jorge L.; Hoffmann H.; Sabadini E. Rheological and calorimetric study of alkytrimethylammonium bromide-sodium salicylate wormlike micelles in aqueous binary systems. J. Colloid and Interf. Sci. 2018, 515, 198-207.

33.

Verwey E. J. W.; Overbeek J. Th. G. Theory of the Stability of Lyophobic Colloids. Elsevier, New York 1948

34.

Hamaker H. C. The London- van der Waals attraction between spherical particles. Physica IV 1937, 10, 1058-1072.

35.

Marcelja S.; Radic N. Repulsion of interfaces due to boundary water. Chem. Phys. Letters 1976, 42 (1), 129-130.

36.

Radic N.; Marcelja S. Solvent contribution to the Debye screening length. Chem. Phys. Letters 1978, 55 (2), 377-379.

37.

Kornyshev A. A.; Kossakowski D. A.; Leikin S. Surface phase transitions and hydration forces. J. Chem. Phys. 1992, 97, 6809-6820.

38.

Gompper G.; Hauser M.; Kornyshev A. A. “Confined" water and hydrophobic attraction as a result of metastable coordination, stabilized by hydrophobic surfaces. J. Chem. Phys. 1994, 101 (4), 3378-3398.

39.

Israelachvili J.; Pashley R. The hydrophobic interaction is long range, decaying exponentially with distance. Nature 1982, 300, 341-342.

40.

Claesson P. M.; Christenson H. Very Long Range Attractive Forces between Uncharged Hydrocarbon and Fluorocarbon Surfaces in Water. J. Chem. Phys. 1988, 92, 1650-1655.

41.

Rau D. C.; Parsegian V. A. Direct measurement of the intermolecular forces between counterion-condensed DNA double helices. Evidence for long range attractive hydration forces. Biophysical Journal 1992, 61, 246-259.

42.

Yoon R-H.; Ravishankar S. A. Long-Range Hydrophobic Forces between Mica Surfaces in Alkaline Dodecylammonium Chloride Solutions. J. Colloid and Interf. Sci. 1996, 179, 403-411. 37 ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 39

43.

Meyer E. E.; Rosenberg K. J.; Israelachvili J. Recent progress in understanding hydrophobic interactions. PNAS 2006, 103, 43, 15739-15746.

44.

Helm, C. A.; Israelachvili, J.; McGuiggan P. M. Role of hydrophobic forces in bilayer adhesion and fusion. Biochemistry 1992, 31, 1794 – 1805.

45.

Leckband D.; Israelachvili J. Intermolecular forces in biology. Quaterly Review of Biophysics 2001, 34, 105-267.

46.

Bernheim-Groswasser A.; Zana R.; Talmon Y. Sphere-to-Cylinder Transition in aqueous micellar solution of a dimeric (gemini) surfactant. J. Phys. Chem. B 2000, 104, 4005-4009.

47.

Schmiedel P., Dissertation Bayreuth 1995

48.

Hammer M. U.; Anderson T. H.; Chaimovich A.; Shell Scott M.; Israelachvili J. The search for the hydrophobic force law. Faraday Discuss. 2010, 146, 299401.

49.

Stuart H. A.; Peterlin A. Optische Anisotropie und Form von Fadenmolekülen. I. Depolarisationsgrad der Lichtzerstreuung von Lösungen mit Fadenmolekülen. J. Polymer Sci. 1949, V, 5, 543-549.

50.

Stuart H. A.; Peterlin A. Optische Anisotropie und Form von Fadenmolekülen. II. Künstliche Doppelbrechung. J. Polymer Sci. 1949, V, 5, 551-563.

51.

Israelachvili J. N. Intermolecular and Surface Forces. Third Ed., Elsevier, New York 2011

52.

Hirschfelder J.O., Curtiss C.F. and Bird R.B. Molecular Theory of Gases and Liquids. John Wiley & Sons, New York 1964, 624-634.

53.

Schroyen B., Swan J.W., Van Puyvelde P. and Vermant J. Quantifying the dispersion quality of partially aggregated colloidal dispersions by high frequency rheology. Soft Matter 2017, 13, 7897-7906.

54.

Willenbacher N., Oelschlaegel C., Schopferer M., Fischer P., Cardinaux F. and Scheffolf F. Broad Bandwidth Optical and Mechanical Rheometry of Wormlike Micelle Solutions. Phys. Rev. Lett.,2007, 99, 068302


Page 39 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

TOC


Evidence of sticky contacts between the wormlike micelles (WLM) in

Recommend Documents