Surfactants Document prepared by Hervé This le 27th December 2010 Oil does not mix with water; surface tension Why do we use soap in order to wash one's hands? Why do we pub soap in a cloth washing machine? The answer is hidden in the title of this document. But experiment is fine: in a vessel, let's put water and oil. You see that oil poured over water sinks first, and then comes up to the surface (do you have an idea of the velocity of the phenomenon? do you know why oil is floating on water?). There is another document for the explanation of this phenomenon, but here I propose a specific way for the description. The rule, when you envision an evolution of the world, it's to calculate free enthalpy (or free energy, depending whether you work at constant pressure, or at constant volume). Let's use this method. Let's assume that the length of the cubic vessel is equal to 1 (u.a. = arbitrary units; we don't care about this). Then, the area of the contact surface is:
A = 1.1 = 1. The world being governed (in particular) by the first and by the second principles of thermodynamics, one can consider that when oil drops dip and then go up are obeying this principles, so that, at equilibrium (very important word, in the context of thermodynamics): dg = 0. G = Gfinal - G initial. ] The world being as it is, G has to decrease in spontaneous transformations (if you need more, please read The second law, by P. W. Atkins, Freeman and Co, NY, USA). And this explains why oil goes up, after dipping (and water, accordingly, goes down): this is described by an evolution such as: G< 0 You remember perhaps that: G = H - T.S. In this very important equation, H, enthalpy, is representing the energy of the system at constant pressure. But this being said, it is true, as oil is less dense than water, that the motion of oil and water is associated to a decrease of enthalpy (potential energy). However, you remember also that interfaces separating two media are the places where forces act. This is demonstrated when you add water to a glass of water, more than the maximum level of the glass: If water can make a layer over the surface of the glass, if it does not flow, it means that forces keep it. On can understand, in particular, that hydrogen forces between water molecules in the liquid are not equilibrated by possible forces (Van der Waals) between water molecules and air molecules. This is exactly what occurs when you put oil with water. Let's analyze more. If forces "group" water molecules, it means that water molecules don't like (in terms of energy) to be at a surface. Otherly stated, free enthalpy depends on the area of the interface.
The simplest possibility would be to say: Gf A. Why such a proportionality? Befauce it is by the interface that hydrogen bonds can act, for example: the bigger the area, the more bonds there are. No reasons to put square or cubic laws. However, instead of writing, shouldn't we write more correctly : surface tension. Let's come back to oil floating over water. Let's divide the oil in two parts, that we put in water: Here the area of the interface oil/water is twice the area of one the blocks. For one oil volume, the lower surface has an area : 0,5.1 Idem for the upper surface. For the face at the left (assuming a thickness equal to 0,5) : 0,5.1 Idem for the face at the right. For the face in front: 0,5.0,5 Idem for behind. Which means, for one volume, a total area equal to : A 1 = 2 (0,5 + 0,5 + 0,5.05). And for the 2 volumes: A = 2 A 1 = 4 (0,5 + 0,5 + 0,5.05) = 5. The area being multiplied by 5, on can admit that the free enthalpy was also multiplied by 5. Let's go on with divisions, but let's change the way we are calculating: now we admit taht we make small cubes of lenght c. The volume of one cube is. Assuming an initial volume V, the number of cubes is equal to. [Remember that liquid cannot be compressed, because they are "condensed matter")
The area of one cube is:. And the total area: (1.1) (1.2) Let's now admit that the initial volume has a length C : (1.3) (1.4) Think that c is a fraction of C : (1.5) The area is: There is an increase! (1.6) As a consequence, one has to give energy to the system in order to disperse oil in water. Surfactants Let us now examine another experiment: starting from the previous state, with water over the level of the glass, let's dip a clean needle in water. The needle is percing the surface without any particular effect. Now let's dip first the needle is liquid soap, and repeat the experiment of piercing the surface: oh! All water runs over. We shall reach the conclusion that the "suface tension" of water was reduced, as this energy is not enough to fight the potential energy which wanted to make water flow downward. By the way, such a modification is helpful when we want to disperse oil in water: if we reduce the surface tension, we also reduce the amount of energy necessary to disperse oil in water. Why did soap had its effect? Soap are made since ancient populations of Gaule (much before Romans won war in the country today called France), by heating fat and wood ashes (did you try it?). The French chemist Michel Eugène Chevreul explained chemically the process, showing in 1823 that potash (KOH, from ashes) could make a "saponification".
But what are fats? Oil, for example, is a mixture of triglycerides: You see: a glycerol residue is linked through ester links to residues R1, R2, R3, which are alkyl chains, saturated or not. In other words, saponification generates glycerol, on one hand, and carboxylate ions, on the other hand. Carboxylate? CH3-CH2-CH2-CH2-CH2-CH2-CH2...-COO - (please don't forget the counter ion K + ). There is a negatively charged end (the "head") and another aliphatic end ("tail"). If one put such molecules in a glass with water and oil, the head+tail molecules is going at the interface such as the hydrophilic head is in water, and the hydrophobic in oil : Let's add more surfactants. You can dissolve some, because an equilibrium is being established, as when you put a liquid and vacuum in a closed vessel: the liquid is being in equilibrium with its vapor pressure :
How can we describe such a distribution? One way to do it is to use the "log P", i.e. the logarithm of the distribution coefficient of compounds between n-octane (do you know what n-octane is? Please investigate if not) and water. Please do learn that even the simplest software of molecular modelling (e.g. ChemDraw, but there are many others, free on line, for Linux) can calculate this, and produce a rough estimation of this value. Let's not forget the physical chemistry that we know already: what was shown in this picture, as purple disks, are molecules. Such molecules are (often, not always) organic compounds, as we could see. Molecules such as alcanes C n H 2n+2, branched or not, are very poorly soluble in water, but soluble in octane (or in oil). This can be explained: a mass of such compounds has few cohesion, because the molecules are linked only by van der Waals forces. Water, on the other hand, is made of molecules linked together by hydrogen force, stronger than van der Waals forces. It can be shown that the introduction an alcane molecule among water molecules can reduce the entropy of the whole, because water molecules cannot move as freely as they would otherwise. But remember that G = H - T S. This means that is S is reduced, T.S is reduced more, and - T.S is increased : G increases, which is not good. Let's create various molecules in silico, and let's calculate their log P, or (it's almost the same, the log Kow, for an octanol/water system) : CH3-OH CH3-CH2-OH -5.54 CH3-CH2-CH2-OH -4.7 CH3-CH2-CH2-CH2-3.86
-OH CH3-CH2-CH2-CH2 -CH2-OH -3.0 CH3-CH2-CH2-CH3 4.58 Phi-OH -3.52 You see: the various compounds have various affinities for the various phases. HLB Going back to surfactants, this scale is interesting, but it is not used. Users of surfactants, in the industry, have another scale more like ph scale: surfactants are ranked between 0 and 14. It's called HLB, or "Hydrophilic Lipophilic Balance": the name is poorly chosen, because there is no "balance", from a thermodynamical point of view. The idea is to start from a oil over water system. Then you add a small quantity of a surfactant. Part of the surfactant molecules go to the oil-water interface, but this part is not much : assuming 1 g of surfactant added to a system made of 100 g of water and 100 g of oil, in a vessel of section 10 cm 2, assuming that the size of the polar heads is 10-9.10-9 m 2 (the size is chosen remembering that a covalent bond is about 10-10 m long), i.e. 10-18 m2, on calculate a number of molecules of the surfactant equal to, i.e.. Let consider for example an alkyl chain with 15 carbon atoms, with hydrophilic head such as COO-, then the molar mass of this surfactant would be about, which is to say about 250 g. In other words, the mass of surfactant at the interface is of the order of, which is about 4.10-4 g. in DSF). This compound distributes so that there is a concentration c[oil] in oil, and a concentration c [water] in water. Le partition coefficient k is equal to : Using this coefficient is very difficult, as for log P,because some compounds are very hydrophilic (ions, for example and also saccharides, and more generally polyols, establishing a lot of hydrogen bonds with water molecules), and, on the other hand, very hydrophobic compounds (compounds with long aliphatic chain). This is why we can make the following reasoning. Let's assume a surfactant molecule that we move from water toward oil. The necessary work W is obeing the general Boltzmann law: Let's calculate W assuming an energy E H for transferring an hydrophilic group, and E L transferring a lipophilic group. Let n H the number of hydrophilic groups, and n L the number of lipophilic groups. Then, in this approached description:
And then: We get : The coefficients and can be measured. In 1946, Griffin proposed to define HLB from a relationship of this kind, changing thinks so that the scale would be between 0 and 14 : = 7 + ( ) Here the a constant is chosen so that the scale is like for ph. The following table is giving the q parameters for some chemical groups: Hydrophilic group Hydrophobic group -CO2Na 19.1 -CH2- -SO3Na 11.0 -N(CH3)3Cl 9.4 -O- 1.3 -OH 1.9 Why using such a scale? You can use it for chosing the needed surfactants, in some particular applications. For example, when you want to make an emulsion, or a foam, or when you want to avoid a foam... Here is it: HLB use 1.5-3 antifoaming agents 3-6 emulsions W/O 7-9 foaming agents, wetting agents 8-18 emulsions O/W 13-15 soaps 15-20 solubilisations organic compounds
Making colloidal systems was leading to the dispersion of these compounds at the oil-water interface, then to the distribution of the compound between the two phases. What we did not see, it is that the surfactants can associate when their concentration is enough. In particular, they can make "micelles" (from Latine mica = part), polar heads being in water, and hydrophobic tail being far from water. The picture shows that micelles are not the only possibilities. What is does not show, it is the MAIN : it's wrong! Indeed, one should never forget that the structures that you see here are molecular assemblies, inside a liquid whose molecules are in motion. One should absolutely remember, when looking to such pictures, and molecules go and move, exchanging between the liquid environment and the structure. Each surfactant molecule can at any time leave the structure, micelle or other one, being replaced by a water molecule or another sufactant molecule, or by nothing. What you see is an "average" picture. It is true, for example, that the residence time of a water molecule in a micelle is very short, so that if a picture was taken, making the average on various micelles, no water would appear.
Why are micelles spherical? The answer is simple: head are much bigger than tails. Indeed you cannot trust the chemical appearance of head and tails, because "steric effects", for example, are only a result of electromagnetic effects. The head are "big" because they are electraclly charged, or have non binding doubles (on oxygen atoms, for example), whereas tails are long, but "thin", as only van der Waals forces apply. On can calculate the structure of micelles, assuming that heads hava a projected area ("cross section") A, and a tail a volume V= (here n stands for the number of groups in the hydrophobic chain, and v is the volume of one group; we assume micelles of radius R. Let N be the average number of molecules of surfactant per micelle. The total volume of the micelle is N.V. This volume can be put in relation with R : The surface, on the other hand, has an area: From these two equations, the radius can be calculated: And the average number of surfactant molecules per micelle: Experimentally the order of magnitude of N is measured to be e100. And now, the calculation of the critical micellar concentration Micelles form only -it was said before- when their concentration is bigger than a particular concentration which is called "critical micellar concentration" (cmc). Can we calculate it? As the world lies in particular on two principles (the first and the second principles of thermodynamics), we have to use the Gibbs function, or free enthalpy, in order to take these two principles into account. And as concentrations are concerned, we have to make a link with energy. Look, concentration c, on one hand, and energy G on the other. The definition of chemical potential is indeed the relation of the two, but instead of having : we have the more general and more exact definition : where i is for the chemical specied i, and n i is the number of moles of this species. Let's consider a surfactant molecule, which can either be isolated ( ), or associated with others ( ). For a charged surfactant: You can see here that the expression is given for a molecule ( ) and not for a mole, as you are used. The factor 2 is because the surfactant is ionic, made from one ion and one counterion.
For the same molecule, in a micelle : Here the entropic term is divided by N, because the micelle behaves as a unique structure with three freedom degrees. As N is large, this term can be dropped, so that the equilibrium between isolated sufactants, and associated surfactants can be written: The second term can be written if we use the results obtained before for HLB. Then : Surfactants for emulsions If surfactant molecules can associate in micelles, they can, also, when an hydrophobic phase is present, lower the surface tension, so that one can make an oil into water emulsion. This is how you can wash your clothes in water, using soap, when it is made dirty with organic matter such as fat. Look at this "film", when a fiber with fat is put in water with soap. Of course, you don't see the soap molecules moving toward the interface, and slowly making a sphere of fat covered with surfactant, but this is indeed the phenomenon :
By the way, I realize now that I forgot to say that emulsions are not stable. Do we calculate? Slightly out of the question, but don't forget: in an emulsion (never stable), small droplets cream slower than big ones Let's consider a droplet of oil of radius r in water. We want to calculate creaming. First, we shall assume that droplets are so big that they are not moved by Brownian motion. 3 forces act on droplets 1. the weight : 2. Archimedes force (buyancy) : (8.1) (8.2) 3. friction, as expressed by the Stokes force : The sum of all forces is equal to the product of the mass by the acceleration (the z axis is chosen to be in the up direction) : (8.3) If you solve this equation, you'll find (please see the big document on creaming and sedimentation) an initial acceleration, and then a motion at (almost) constant velocity. How much is this velocity? We find it by writing that at that time the acceleration is nil: Or : You feel that the velocity is inversely proportional to the radius... but mind that the radius lies in masses! Indeed: And then : Here we are: you see that the velocity is increasing with the square of the radius! In other words, small droplets are slower than big ones.