micelle, a1
TRANSCRIPT
MICELLES Physical Chemistry Lab Course
Assitant: Aravind Kumar Chandiran
Fedora Bonomi-‐Karkour & Alicia Solano
(Group A1)
7th March 2013
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1. AIM The aim of this manipulation is to bring out the formation of surfactant aggregates and to estimate the critical micellar concentration CMC of tensioactive agents. Three different methods will be employed in order to determine the CMC: the measurement of the surface tension, of the conductivity and of the absorption in function of the surfactant concentration. The uncertainty for each result will also be calculated. 2. THEORECTICAL BACKGROUND
2.1. Surfactants
Surface-‐active agents or more simply called surfactants are amphipathic molecules. They are composed of a hydrophobic tail, generally an alkyl chain, and a hydrophilic head, which can be neutral, cationic, anionic or zwitterionic. In an aqueous solution, at low concentration, monomer surfactants can be found dissolved in the bulk or at the interface. The first option is not energetically favourable due to the hydrophobic effect: a water cage surrounds the surfactant tail. Hence, in order to minimize this effect, the monomers tend to form a monolayer at the air/water interface: the hydrophilic head remains in water, whereas the hydrophobic tail is placed vertically outside the solution (A, B, figure 1). Above a threshold surfactant concentration, surfactants added to the solution start to self-‐aggregate into micelles (C, figure 1), while the monomers concentration remains constant.
Figure 1. Self-‐Assembly mechanism of micelles
A typical micelle, in aqueous solution, has the surfactant hydrophilic heads in contact with the water molecules, sequestering he hydrophobic tails in the micelle center so as to decrease the unfavourable interaction with the solvent. The micelles formation modifies some properties of the solution such as the surface tension, the conductivity or even the turbidity. Hence, the CMC can be determined following these properties in function of the surfactant concentration. It should appear like a break in the tendency. Above the CMC, micelles are formed spontaneously (ΔG=ΔH-‐ΔS<0). But as the absolute value of the micellisation heat is relatively small (ΔH), the aggregation is mainly an entropic effect (ΔS). However, two opposing forces are present. First, the agglomeration of surfactants into micelles reduces the entropy whereas during the micellisation, the solvatation cage surrounding the hydrophobic tails is destructed and that leads to an entropy increase. This last component has a higher value and then gives a positive entropic balance when micelles are formed. Hence, in water, the hydrophobic effect is the driving force for the micelle formation.
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2.2. Surface tension In the bulk, the intermolecular forces acting on a molecule (i.e. water) compensate each other. But, at the interface air/liquid, some neighbouring molecules are missing. This produces a resulting force F pointing towards the solution (figure 2). Therefore, so as to expand the surface, a work must be done against the force F. For this reason, the surface tension γ is defined as the amount of force necessary to expand the surface of a liquid by one unit (eq. 1).
γ =𝑑𝑤𝑑𝐴
=𝐹𝑥 (𝟏)
Where 𝛾 is the surface tension [N.m-‐1], F is the force [N] and x the length of the interface [m].
Thus, it is possible to determine the surface tension just by measuring the force required to enlarge a surface of known size.
Figure 2. Origin of the surface tension.
The influence of a solute on the surface tension can be understood through the surface excess Γi. For a compound i, it is defined as:
Γ! =𝑛!!
𝐴 (𝟐)
Where 𝛤! is the surface excess [mol.m-‐2], 𝑛!! is the number of moles of i at the surface [mol] and A is the surface area [m2].
The surface excess of a solute can be determined experimentally by plotting the surface tension versus the logarithm10 of the solute concentration (eq. 4). The slope is then proportional to the surface excess.
𝛤𝑏+ = 𝛤𝑏− = −1
2 · 2,303 · 𝑅𝑇·
𝑑𝜎𝑑𝑙𝑜𝑔10𝑐𝑏
𝟑
↔ 𝑑𝜎 = − 2 · 2,303 · 𝑅𝑇𝜞𝒃 · 𝑑𝑙𝑜𝑔!"𝑐! 𝟒
From equation 4, it can be concluded that if the quantity of solute increases at the surface, positive surface excess, then the surface tension is reduced. More intuitively, it can be thought that by adding surfactants to water, they spread over the surface due to the repulsive interaction of their heads. That implies that some of the water molecules are no longer in contact with the air because they have been replaced by the surfactants: this breaks the surface tension. Moreover, the surface energy is decreased by the favourable interaction of the hydrophobic tails, which stabilizes the surface.
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Therefore, before the CMC, the surface tension should decrease with the addition of surfactants. But once at the CMC, as the surface is saturated with monomeric surfactants, and all additional surfactants self-‐assemble, the surface tension should stay constant. Finally, knowing the surface excess allows to determine the area occupied by one surfactant molecule at the surface using the following equation:
𝑎!""!#$%" =1
𝛤! · 𝑁!"#$%&'# (𝟓)
Where a!""!#$%"is the area [m2], and N!"#$%&'# is the Avogadro constant equal to 6.022 ·∙ 1023 mol-‐1
2.3. Conductimetry The conductivity is the ability of a solution to conduct an electric current in a given volume. For measuring it, an alternating current is applied between two electrodes immersed in a solution and then, the resulting voltage is determined. During this process, the cations migrate to the negative electrode whereas the anions move to the positive one. Hence, the conductivity depends on the ions mobility and on their concentration: a higher concentration of free ions in solution leads to an increase of the conductivity. Below the CMC, when surfactants are added to the solution, two ions are liberated: the surfactant head and the counterion. Therefore, due to the increase of charge carriers in solution, the conductivity will sharply increase. However, above the CMC, monomers will start to aggregate in micelles. Since these structures are much larger than a surfactant monomer, they diffuse more slowly through the solution and hence their mobility is reduced. Therefore, after the CMC, the addition of surfactants should still increase the conductivity but more slowly.
2.4. Spectroscopy
Coumarine-‐6 (figure 3) is a non-‐polar compound. Then, according to the rule “like dissolves like” this molecule will not be soluble in a polar solvent like pure water. However, a strategy to dissolve this compound in water is if it is placed in a micelle core. In that case, the molecule will only interact with the surfactants hydrophobic tails, whereas the heads will solubilize the micelle in the bulk.
Figure 3. Structure of Coumarine-‐6 dye.
Hence, before the CMC, the amount of coumarine-‐6 solubilized should be almost zero. Whereas, above the CMC, the solubilization should start and increase as more surfactants are added to the solution. The amount of coumarine-‐6 solubilized in the solution can be determined using UV-‐VIS spectroscopy and applying the Beer-‐Lambert law (equation 1). It has a maximal absorption for a wavelength of 472nm. 𝑨 = 𝐥𝐨𝐠 𝑰𝒐
𝑰= 𝜺 ∗ 𝑪 ∗ 𝒍 (1)
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Where A is the absorbance, 𝐼! and I are the intensities of the incident and outgoing light over the sample, 𝜀 is the molar extinction coefficient [L·∙mol−1·∙cm−1], C is the molar concentration of the dye in the solution [mol.L−1] and l is the cuvette length [cm].
Moreover, knowing the micelle concentration in the solution, the number of coumarine-‐6 molecules inside one micelle can be calculated (equation 2). 𝑴 = 𝑺 !𝑪𝑴𝑪
𝒗 (2)
Where [M] and [S] are respectively the micelle and surfactant concentrations [mol.L−1], CMC is the critical micelle concentration [mol.L−1] and v is the aggregation number.
3. EXPERIMENTAL SETUP
3.1. Solutions preparation Two surfactants are used during this work. CTAB (CetylTrimethylAmmonium Bromide) is a cationic surfactant whereas NALS (Lauryl Sulfate of Sodium) is an anionic one (figure 1).
A B Figure 4. (A) CTAB surfactant. (B) NALS surfactant.
So as to set up a concentration gradient (table 1), 15 solutions are prepared for each surfactant. A certain volume is pipetted from a stock solution, and completed to 10 mL with water.
NALS Concentration [mM]
CTAB Concentration [mM]
1 0.1 2 0.2 3 0.3 4 0.4 5 0.5 7 0.7 8 0.8 9 0.9 10 1 20 2 35 3.5 50 5 65 6.5 80 8 100 10
Table 1. Surfactant concentration in each solution.
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3.2. Surface tension
The surface tension γ of each solution is measured three times using a Du Noüy ring tensiometer. More precisely, this apparatus determines the force required for a platinum ring to detach from a liquid surface, that is to say, to overcome the surface tension. First, the solution is placed into a suitable beaker. Then, the ring is fully submerged into the solution (1 to 4, figure 2) and then lifted until it is placed at the liquid-‐air interface (5, figure 2). Subsequently, the main dial is set to zero and the black-‐and-‐white circle is calibrated. Finally, the ring is slowly raised upwards (6 to 8, figure 2) by turning the main dial until the water layer breaks. The main dial shows directly the surface tension value in mN/m. Between two different solution measurements, the beaker and the ring are cleaned with deionized water and then dried with a tissue.
Figure 5. Steps to measure the surface tension using a Du Noüy tensiometer.
3.3. Conductimetry
Before any measurement, some initial settings have to be applied on the device. The conductivity meter, with a cell constant of 0,8 cm-‐1, needs to be calibrated with a potassium chloride standard solution. And, since the temperature strongly influences the conductivity of a solution, the automatic temperature calibration has to be switched on. Then, the conductivities are measured three times for each of the 15 solutions prepared for the two surfactants. During the measurement, the cell must be straight and completely submerged into the solution. Between two different solutions, the cell is cleaned with distilled water and then dried with an absorbent tissue.
3.4. Non-‐polar compound
One week before, some coumarine-‐6 orange crystals were added to the last group solutions since the equilibrium between solid and solubilized coumarine-‐6 takes several days to settle.
Before running the solution measurements, the UV-‐VIS spectrometer must be calibrated. First, the spectrometer cuvette is filled with the most concentrated solution and its absorbance is measured. Some parameters are settled: the integration time (2ms), the scans to average (50) and the boxcar width (20). The higher the integration time the longer the detector “looks” at the incoming photons. The scan average allows to average consecutive scans so as to smooth out noise. And the boxcar width is the average of the counts from adjacent group of pixels, so as to get a smoother spectrum. Next, a dark reading is made by blocking with a cover any light that could enter the spectrometer. Finally, after changing the cuvette by one
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filled only with water, a blank measurement is also done. This calibration needs to be performed two times, one for each surfactant set of solutions.
After that, the absorbance for each solution is measured at 471,8 nm three times with an interval of 5 seconds. This wavelength corresponds to the maximal absorption of coumarine-‐6.
Between two different solutions, the cuvette is rinse with deionized water.
4. EXPERIMENTAL RESULTS
4.1. Surface tension The CMC is estimated by plotting the surface tension versus the logarithm of the surfactant concentration.
• NALS Surfactant
Figure 6. Surface tension in function of log [NALS].
The two linear regressions together with the error calculations are determine following the procedures located in the annexes 4 and 5. The technique is more widely explained in the reference 1. Thus, the resulting equations are:
Before the CMC: 𝜎 = −20.77 ±1.49 ∙ 𝑙𝑜𝑔 𝑁𝐴𝐿𝑆 + (57.25 ± 0.93)
After the CMC: 𝜎 = −2.60 ±0.78 ∙ 𝑙𝑜𝑔 𝑁𝐴𝐿𝑆 + (41.25 ± 1.23)
y = -‐20,77x + 57,25 R² = 0,97
y = -‐2,60x + 41,25 R² = 0,65
0
10
20
30
40
50
60
70
0 0,5 1 1,5 2 2,5
Surface Tension / [m
N/m
]
log [NALS]
Before CMC
Aker CMC
Lineal (Before CMC)
Lineal (Aker CMC)
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The CMC is placed at the intersection of the two linear regressions. Then, it is equal to:
CMCNALS=8.12 ± 0,52 mmol.L-‐1 The superficial excess before the CMC at T=293K, can be calculated using equation 3:
𝚪𝒃 = −𝟏
𝟐.𝟑𝟎𝟑 ∙ 𝟖.𝟑𝟏𝟒 ∙ 𝟐𝟗𝟑−𝟐𝟎.𝟕𝟕 ±𝟏.𝟒𝟗 ∙ 𝟏𝟎!𝟑 = 𝟑.𝟕𝟎 ± 𝟏.𝟒𝟗 ∙ 𝟏𝟎!𝟔 𝒎𝒐𝒍.𝒎!𝟐
And the effective area occupied by a molecule of surfactant is determined using equation 5:
𝑨𝒆𝒇𝒇 =𝟏
𝟑.𝟕𝟎 ±𝟏.𝟒𝟗 ∙ 𝟏𝟎!𝟔 ∙ 𝟔.𝟎𝟐 ∙ 𝟏𝟎𝟐𝟑= (𝟒.𝟒𝟗 ± 𝟎,𝟓𝟔) ∙ 𝟏𝟎!𝟏𝟗 𝒎𝟐
• CTAB Surfactant
Figure 7. Surface tension in function of log [CTAB].
The two linear regressions are:
Before the CMC: 𝜎 = −15.84 ±3.89 ∙ 𝑙𝑜𝑔 𝐶𝑇𝐴𝐵 + (42.89 ± 2.26)
After the CMC: 𝜎 = −4.55 ±0.29 ∙ 𝑙𝑜𝑔 𝐶𝑇𝐴𝐵 + (37.59 ± 0.19)
y = -‐20,95x + 39,38 R² = 0,85
y = -‐4,0338x + 37,193 R² = 0,96562
0
10
20
30
40
50
60
70
-‐1,5 -‐1 -‐0,5 0 0,5 1 1,5
Surface Tension / [m
N/m
]
log [CTAB]
Before CMC Aker CMC Lineal (Before CMC) Lineal (Aker CMC)
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From these two regressions and equations 3 and 5, the following results have been calculated.
CMCCTAB=2,75 ± 1,14mmol.L-‐1
𝚪𝐛 = 𝟐.𝟖𝟐 ± 𝟑.𝟖𝟗 ∙ 𝟏𝟎!𝟔 𝐦𝐨𝐥/𝐦𝟐 𝐀𝐞𝐟𝐟 = (𝟓.𝟖𝟗 ± 𝟎,𝟗𝟓) ∙ 𝟏𝟎!𝟏𝟗 𝐦𝟐
• Discussion
As expected, the surface tension diminishes linearly before the CMC when the surfactant concentration increases. But after the CMC, the conductivity stays approximately constant. The reasons of this trend have been explain in the theoretical part. The effective areas obtained, can be compared with the theoretical values. Only the results are shown in table 2, the calculations for the theoretical effective area are placed in the annex 6.
Surfactant 𝐀𝐞𝐟𝐟,𝐞𝐱𝐩 [𝐦𝟐] 𝐀𝐞𝐟𝐟,𝐭𝐡 [𝐦𝟐] NALS (4.49 ± 0,56) ∙ 10!!" 1,064!!" CTAB (5.89 ± 0,95) ∙ 10!!" 6,807.10!!"
Table 2. Comparative table between the experimental and theoretical values for the effective area. Some conclusions can me made looking at table 2. First, the experimental values are for the two surfactants higher than the theoretical. This implies, that the effective area occupied by a hydrophilic head at the surface is bigger than the disk used to describe it. Moreover, even if the error for NALS is very large (322%), the experimental and theoretical values have the same magnitude. The same idea applies to CTAB. Finally, as expected, the area occupied by a sulphate (NALS) is lower than the area occupied by a quaternary amine.
4.2. Conductimetry
In order to determine the CMC for each surfactant, the conductivity is plotted versus the surfactant concentration.
• NALS Surfactant
Figure 8. Conductivity in function NALS concentration.
y = 42,68x + 31,63 R² = 0,9938
y = 17,72x + 238,9 R² = 0,99829
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120
Cond
uc\v
ity / [μ
S.cm
-‐1 ]
NALS Concentra\on / [mmol.L-‐1 ]
Before CMC Aker CMC Linear regression before CMC Linear regression aker CMC
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The two linear regressions calculated are:
Before the CMC: 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 42,7 ± 1,51 𝑁𝐴𝐿𝑆 + (31,6 ± 7,39)
After the CMC: 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 17,7 ± 0,26 𝑁𝐴𝐿𝑆 + (238 ± 15,8) And again, the CMC is placed at the intersection of the two linear regressions:
CMCNALS=8,31 ± 0,86 mmol.L-‐1
• CTAB Surfactant
Figure 9. Conductivity in function CTAB concentration.
The two linear regressions calculated are:
Before the CMC: 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 73,3 ± 7,31 · 𝑁𝐴𝐿𝑆 + (4,13 ± 2,16)
After the CMC: 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 15,5 ± 0,33 · 𝑁𝐴𝐿𝑆 + (53,5 ± 1,99) And again, the CMC is placed at the intersection of the two linear regressions:
CMCCTAB= 0,86 ± 0,12 mmol.L-‐1
• Discussion Again, the hypothesis exposed about the conductivity trend in the theoretical part is demonstrated by the experimentation. As expected, the curve slope before the CMC is bigger than the curve slope after the CMC.
y = 73,3x + 4,13 R² = 0,98226
y = 15,5x + 53,5 R² = 0,99769
0,00
50,00
100,00
150,00
200,00
250,00
0,00 2,00 4,00 6,00 8,00 10,00 12,00
Cond
uc\v
ity / [μ
S/cm
]
CTAB Concentra\on / [mmol/L]
Before CMC
Aker CMC
Linear regression before CMC
Linear regression aker CMC
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4.3. Non-‐polar compound
• NALS Surfactant
Figure 10. Coumarine-‐6 concentration in function of log[NALS]
The two linear regressions calculated are:
Before the CMC: 𝐶𝑜𝑢𝑚𝑎𝑡𝑖𝑛𝑒 − 6 = −1,91 ± 0,55 · 10!! · 𝑙𝑜𝑔 𝑁𝐴𝐿𝑆 + 4,46 ± 0,26 · 10!!
After the CMC: 𝐶𝑜𝑢𝑚𝑎𝑡𝑖𝑛𝑒 − 6 = 1,06 ± 0,12 · 10!! · 𝑙𝑜𝑔 𝑁𝐴𝐿𝑆 + −9,05 ± 1,78 · 10!! And again, the CMC is placed at the intersection of the two linear regressions:
CMCNALS= 7,65 ± 0,64mmol.L-‐1
y = 1,06.10-‐2 x -‐ 9,05.10-‐3 R² = 0,90495
y = -‐1,91.10-‐4x + 4,46.10-‐4 R² = 0,80059
0,00
0,00
0,00
0,01
0,01
0,01
0,01
0,01
0,02
0,00 0,50 1,00 1,50 2,00 2,50
[Coumarine-‐6] / [m
mol/L]
log [NALS]
Before CMC After CMC Lineal (After CMC) Linear regression before CMC
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Figure 11. Coumarine-‐6 concentration in function of Micelle concentration for NALS
The regression curve is:
𝐶𝑜𝑢𝑚𝑎𝑡𝑖𝑛𝑒 − 6 = 1,17 ± 0,42 · 10!! · [𝑀𝑖𝑐𝑒𝑙𝑙𝑒𝑠] + 1,01 ± 0,76 · 10!! Therefore, in a micelle composed by 100 NALS surfactants, there is only one (1,17 rounded to 1 molecule) molecule of coumarine-‐6 placed inside the micelle.
• CTAB Surfactant
Figure 12. Coumarine-‐6 concentration in function of log[CTAB]
y = 0,0117x + 0,001 R² = 0,97733
0,00
0,00
0,00
0,01
0,01
0,01
0,01
0,01
0,02
0,00 0,20 0,40 0,60 0,80 1,00 1,20 1,40
[Cou
marine-‐6] / [m
mol/L]
[Micelles] / [mmol/L]
Serie2 Linear regression
y = 1E-‐06x + 9E-‐05 R² = 0,0002
y = 0,0017x + 0,0008 R² = 0,88738
0,000
0,001
0,001
0,002
0,002
0,003
0,003
-‐1,50 -‐1,00 -‐0,50 0,00 0,50 1,00 1,50
[Cou
marine-‐6] / [m
ol/L]
log [CTAB]
Before CMC Aker CMC Linear regression before CMC Linear regression aker CMC
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The two linear regressions calculated are:
Before the CMC: 𝐶𝑜𝑢𝑚𝑎𝑡𝑖𝑛𝑒 − 6 = 1,31 ± 4,13 · 10!! · 𝑙𝑜𝑔 𝐶𝑇𝐴𝐵 + 8,99 ± 4,60 · 10!!
After the CMC: 𝐶𝑜𝑢𝑚𝑎𝑡𝑖𝑛𝑒 − 6 = 1,74 ± 0,25 10!! · 𝑙𝑜𝑔 𝐶𝑇𝐴𝐵 + 7,86 ± 1,64 · 10!! And again, the CMC is placed at the intersection of the two linear regressions:
CMCCTAB= 0,40 ± 0,25mmol.L-‐1
Figure 13. Coumarine-‐6 concentration in function of Micelle concentration for CTAB
The linear regression curve is:
𝐶𝑜𝑢𝑚𝑎𝑡𝑖𝑛𝑒 − 6 = 2,10 ± 0,40 · 10!! · [𝑀𝑖𝑐𝑒𝑙𝑙𝑒𝑠] + 5,82 ± 6,23 · 10!! Therefore, in a micelle composed by 100 NALS surfactants, there are two (2,10 rounded to 2 molecule) molecules of coumarine-‐6 placed inside the micelle.
• Discussion As expected, the solubilisation of Coumarine-‐6 only starts after the CMC. Hence, before the CMC, the absorbance is almost zero. That implies that there is no dye solubilized in the solution. Only after the CMC, the absorbance and hence the dye concentration increases linearly when the surfactant concentration increase. Figure 11 and 12 don’t show a linear trend. Instead, they seem to follow a logarithmic trend. The reason for this is can only come from handling errors during the experiment.
y = 0,0007ln(x) + 0,0041 R² = 0,94973
y = 0,021x + 0,0006 R² = 0,70899
0,0000
0,0005
0,0010
0,0015
0,0020
0,0025
0,0030
0,00 0,02 0,04 0,06 0,08 0,10 0,12
[Cou
marine-‐6] / [m
ol/L]
[Micelle] / [mmol/L]
Data set Logarithmic regression Lineal regression
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5. METHODS COMPARISON AND SOURCE ERRORS
CMC [mM] (relative error %) Surface tension Conductivity Non-‐polar compound Theoretical value
NALS 8.12 ± 0,52 (14,5%) 8,31 ± 0,86 (12,5%) 7,65 ± 0,64 (19,4%) 9,5
CTAB 2,75 ± 1,14 (198%) 0,86 ± 0,12(0,6%) 0,40 ± 0,25 (5%) 0,92
With method giving the lowest relative error is the conductivity. The surface tension seems to give better results than the solubility method. However, the relative error for the CTAB CMC with the surface tension method is extremely large (198%) and should not be taken into account. The common error source for the three methods comes mainly from the manipulation error (i.e. Solution preparation) and from the instrumental errors (i.e. errors in the pipette volumes). Moreover, the temperature changes can affect the measures. The surface tension method requires the platinum ring to be in a perfect state. That was not the case during this practical work: the ring was totally ben don one side. Moreover, the presence of bubbles could also affect the measures. To obtain accurate results with the conductivity method, the electrodes must be perpendicular over the surface. Besides, it could be a good idea to agitate the solution to obtain a homogenous one, Finally, for the last method, some crystals were present in the CTAB solution. The probably have affected the absorption measurements. 6. CONCLUSION The CMC of two different surfactants, NALS and CTAB, has been determined through three different techniques: the surface tension, the conductivity and the solubilisation of an non-‐polar compound. Moreover, some other properties like the effective area occupied by the hydrophilic head of a surfactant and the quantity of non-‐polar molecules placed inside one micelle after the CMC have been calculated. The conductivity method gives the most accurate results and is also the faster one. Whereas the superficial tension method takes a lot of time. The CMC could also have been determined using other solution properties like the osmotic pressure and the turbidity.
7. ANNEXES
7.1 Surface tension method: experimental data
[NALS] [mM] Average surface tension [mN.m-‐1]
1 57,37 2 50,33 3 46,83 4 46,73 5 42,93 7 38,57 8 37,33 9 38,60 10 38,40 20 37,90 35 37,07 50 38,07 65 37,57 80 35,50 100 35,17
[CTAB] [mM] Average surface tension [mN.m-‐1]
0,1 56,27 0,2 55,27 0,3 54,40 0,4 49,67 0,5 48,13 0,7 42,33 0,8 38,43 0,9 37,97 1 37,33 2 35,77 3,5 34,77 5 34,87 6,5 33,67 8 33,67 10 33,10
7.2 Conductivity method: experimental data
[NALS] [mM] Average conductivity [μS.cm-‐1]
1 76,67 2 116,67 3 170,00 4 188,33 5 238,33 7 340,00 8 371,67 9 420,00 10 430,00 20 570,00 35 840,00 50 1150,00 65 1366,67 80 1633,33 100 2041,67
[CTAB] [mM] Average conductivity [μS.cm-‐1]
0,1 8,15 0,2 21,50 0,3 24,67 0,4 35,00 0,5 44,00 0,7 53,33 0,8 62,00 0,9 65,33 1 67,33 2 84,67 3,5 110,00 5 128,33 6,5 158,33 8 178,33 10 206,67
7.3 Non-‐polar compound method: experimental data
[NALS] [mM] Average Absorption [Coumarine-‐6] [mM] [Micelle] [mM] 1 2,57·∙10-‐2 4,58·∙10-‐4 -‐6,63·∙10-‐2 2 2,13·∙10-‐2 3,81·∙10-‐4 -‐5,38·∙10-‐2 3 1,80·∙10-‐2 3,21·∙10-‐4 -‐4,13·∙10-‐2 4 2,07·∙10-‐2 3,69·∙10-‐4 -‐2,88·∙10-‐2 5 1,73·∙10-‐2 3,10·∙10-‐4 -‐1,63·∙10-‐2 7 4,30·∙10-‐2 7,68·∙10-‐4 8,75·∙10-‐3 8 8,57·∙10-‐2 1,53·∙10-‐3 2,13·∙10-‐2 9 7,57·∙10-‐2 1,35·∙10-‐3 3,38·∙10-‐2 10 5,67·∙10-‐2 1,01·∙10-‐3 4,63·∙10-‐2 20 2,47·∙10-‐1 4,40·∙10-‐3 1,71·∙10-‐1 35 2,12·∙10-‐1 3,79·∙10-‐3 3,59·∙10-‐1 50 4,33·∙10-‐1 7,74·∙10-‐3 5,46·∙10-‐1 65 5,70·∙10-‐1 1,02·∙10-‐2 7,34·∙10-‐1 80 6,65·∙10-‐1 1,19·∙10-‐2 9,21·∙10-‐1 100 8,00·∙10-‐1 1,43·∙10-‐2 1,17
[CTAB] [mM] Average Absorption [Coumarine-‐6] [mM] [Micelle] [mM] 0,1 5,33·∙10-‐3 9,52·∙10-‐5 -‐3,33·∙10-‐3 0,2 6,67·∙10-‐3 1,19·∙10-‐4 -‐2,22·∙10-‐3 0,3 4,00·∙10-‐3 7,14·∙10-‐5 -‐1,11·∙10-‐3 0,4 3,00·∙10-‐3 5,36·∙10-‐5 0 0,5 3,33·∙10-‐3 5,95·∙10-‐5 1,11·∙10-‐3 0,7 5,33·∙10-‐3 9,52·∙10-‐5 3,33·∙10-‐3 0,8 7,33·∙10-‐3 1,31·∙10-‐4 4,44·∙10-‐3 0,9 2,87·∙10-‐2 5,12·∙10-‐4 5,56·∙10-‐3 1 3,07·∙10-‐2 5,48·∙10-‐4 6,67·∙10-‐3 2 9,83·∙10-‐2 1,76·∙10-‐3 1,78·∙10-‐2 3,5 1,10·∙10-‐1 1,97·∙10-‐3 3,44·∙10-‐2 5 1,19·∙10-‐1 2,12·∙10-‐3 5,11·∙10-‐2 6,5 1,22·∙10-‐1 2,18·∙10-‐3 6,78·∙10-‐2 8 1,25·∙10-‐1 2,23·∙10-‐3 8,44·∙10-‐2 10 1,28·∙10-‐1 2,29·∙10-‐3 1,07·∙10-‐1
7.5 Linear Regression For a linear relation between two variables:
y = a + bx Then, the constants a and b are given by:
17
18
7.5 Error analysis
• CMC The CMC is calculated with:
CMC =10a2−a1b1−b2
"
#$
%
&'
Where a2, a1 are the y-‐intercepts and b1, b2 are the curve slopes. Obviously, the index 1 refers to the regression before the CMC, whereas the index 2 refers to the regression after the CMC.
It can be set x equal :
a2 − a1b1 + b2
= x
Thus, the standard deviation over the CMC is determined by:
€
δCMC( )2 =∂CMC∂x
$
% &
'
( ) 2
δx( )2 = 10x ln10( )2δx( )2
δx( )2 = ∂x∂A!
"#
$
%&2
δA( )2 + ∂x∂B!
"#
$
%&2
(δB)2 = 1B!
"#
$
%&2
δA( )2 + −AB2
!
"#
$
%&2
δB( )2
Where :
A=a2-‐a1 and B=b1+b2
And,
δA( )2 = δa2( )2 + δa1( )2
δB( )2 = δb2( )2 + δb1( )2
• Surface excess
€
δΓ( )2 =∂Γ
∂ dσ /d logCsurf( )&
' ( (
)
* + +
2
⋅ δ dσ /d logCsurf( )( )2
= −1
2 ⋅ 2.303RT&
' (
)
* + 2
⋅ δslope( )2
• Surfactant effective area
€
∂aeff( )2
=∂aeff∂Γ
$
% &
'
( ) ⋅ δΓ( )2 = −
1Γ2N
$
% &
'
( ) 2
⋅ δΓ( )2
7.6 Theoretical effective area If we consider the hydrophilic head of a surfactant to occupy an area simplified by a disc, the effective area can be calculated using bond lengths and angles.
• NALS
19
Knowing that:
𝑟!" = 1.485 Å 𝑟!! = 1.066 Å
Thus,
𝐴!"",!! = 𝜋. 𝑟!" + cos 180º − 109,5º · 𝑟!" ! = 1,064!!"𝑚! • CTAB
Knowing that:
𝑟!"= 1.472 Å
Thus, 𝐴!"",!! = 𝜋. 𝑟! = 𝟔,𝟖𝟎𝟕.𝟏𝟎!𝟐𝟎𝒎𝟐