Surface Tension of Polymers Abhinandan Agrawal Hatsopoulos Microfluids Laboratory Department of Mechanical Engineering Massachusetts Institute of Technology 1 June 29, 2005 Physical origin of surface tension/surface energy • “Unhappy” molecules at the surface: they are missing half their attractive interactions • Unbalanced forces for the molecules at the surface lead to additional energy • The additional free energy at the surface is Examples of “minimal surfaces” known as surface energy • This is the fundamental reason behind liquids adjusting their shapes to expose the smallest possible area www.fairfied.edu spherical shape2 June 29, 2005 full dry hair vs. sticky wet hair Jose Bico, Nature, 2004 of bubbles Mechanical definition: as a surface energy Supply of energy is necessary to create surfaces dW =⋅γ dA γ = specific free energy or surface tension (mJ/m2) Specific surface energy of a material is the excess energy per unit area due to the existence of the free surface 3 June 29, 2005 Estimation of surface tension based on intermolecular forces U/2 U is the cohesion energy per molecule inside the liquid U Energy shortfall for a molecule sitting at the surface ~ U/2 a is the size of a molecule; a2 is the exposed area of a molecule U Surface tension is then of order γ ≅ For van der Waals type interactions, Uk≅ T 2a2 At a temperature of 25 oC, kT is equal to 1/40 eV, which gives γ = 20 mJ /m2 (close to actual value for oils and alcohols) Surface tension of a few common liquids in mJ/m2 Helium Ethanol Acetone Glycerol Water Water Molten Mercury (4 K) (100oC) glass 0.1 23 24 63 73 58 ~300 485 U ≈1eV 4 June 29, 2005 hydrogen bonding strongly cohesive liquid Mechanical definition: as a capillary force Surface tension (γ) can also be viewed as a force per unit length (mN/m or N/m) The term “surface tension” is tied to the concept that the surface stays under a tension Examples where surface tension manifests itself as force 1) Slider 2) Capillary adhesion 3) Objects on water (F1+F2)cosθ θ F F2 1 W F dx = 2γ l dx F end view of the leg F1=F2=γl 5 June 29, 2005 Contents Relationship of surface tension with other material properties Derivation of surface tension theoretically from knowledge of intermolecular forces 1) Surface energy of polymer liquids and melts 2) Surface energy of solid polymers 3) Interfacial tension between a solid and a liquid 6 June 29, 2005 1. Surface energy of liquids and melts Existing methods for determining the surface tension of liquids F 1) Wilhelmy’s method, in which one dips a thin plate or p is the perimeter of the plate ring in the liquid and measures the capillary force acting =2(length + width) on the plate θ pγ cos(θ ) 2) The rise of a liquid in a small capillary tube h θ hgρ r γ = 2cosθ r 3) The method of drops, in which one characterizes washing-liquid water the shape of drops r 1 C =− zz + γC = ρgz 223/2 1/2 r ()11++rrzz()r 7 June 29, 2005 z Estimation of surface tension from related properties Since the surface tension is a manifestation of intermolecular forces, it may be expected to be related to other properties derived from intermolecular forces, such as work of cohesion internal pressure, and compressibility. 1) Relationship between work of cohesion and surface tension (Grunberg 1949) N 2/3 V molecular vol.; ⎛⎞A V = molar vol.; N = ⎜⎟= molecules/unit surf. area; A ⎝⎠V 1/3 2/3 γ = surf. energy per unit area; γVN2/3 / 2/3 = surface energy per molecule; WNcoh = 2γ A V A 1/3 2/3 1/3 2/3 γ NVA = molar surf. energy; 2γ NVA = work of cohesion; 2) Relationship between surface tension and solubility parameter (Hildebrand 1950) 1/3 2 0.43 γ /VV∼ molar surf. energy/ =∆E/V=δ ; ⎛⎞γ v δ = 4.1⎜⎟1/3 (cgsunits) ⎝⎠V ∆Ev is the energy of vaporization; δ is the solubility parameter; 3) Relationship between compressibility and surface tension (McGowan 1967) 8 3/2 −8 June 29, 2005 κγ =×1.33 10 (cgsunits) κ ↑→↓γ Introduction to parachor Macleod et al. (Trans. Faraday So., 1923, 19, 38) 4 γ is the surface tension, D and d are γ =−CD()d the density of liquid and vapor at same temperature. C is a a characteristic valid for different compounds constant for a given liquid. and at different temperatures 3 2 1/4 Sugden S. J. (Chem. Soc., 1924, 125, 1177) Unit of Ps (cm /mol)×(erg/cm ) or (m3/mol)×(mJ/m2)1/4 M PC==1/4M γ 1/4 at low temperature, s Dd− d becomes small molecular volume V Ps for different substances is a comparison of molecular volumes at temperatures at which liquids have the same surface tension P bears an constant ratio to the critical volume, which PVs = 0.78 c s suggests that it is a true measure of the molecular volume Ps is called parachor (from παρα = by the side of, and χορος = space to signify 9 comparativeJune 29, 2005 volumes) Useful properties of parachor Ps is only a function of chemical composition and it is an additive function From experimental data for Ps, it is found that X R • Ps can be reproduced by adding together two sets of constants, one for the atoms in the molecule, the other for the unsaturation or ring closure. H C C • The values for a particular atom is independent of the O R1 manner in which it is situated • The values for individual element are same from compound to compound. The molecular parachor is a useful means 4 ⎛⎞Ps of estimating surface tensions γ = ⎜⎟ ⎝⎠V M is the molecular weight, V is the molar volume, ρ is the density. If the group contributions of Ps and V are known, γ results from the above expression 10 June 29, 2005 Atomic and structural contributions to the parachor Deduction of atomic constants: Difference for CH2 in paraffins, esters, ethers, and ketones has a mean value of 40.0 Subtracting nCH2 from a series of values for the paraffins CnH2n+2, values for 2H can be calculated Unit Value Unit Value CH2 40.0 Br 68.0 C9.0 I 90.3 H 15.5 Double bond 16.3–19.1 O 19.8 Triple bond 40.6 O2 (in esters) 54.8 Three-membered ring 12.5 N 17.5 Four-membered ring 6.0 S 49.1 Five-membered ring 3.0 F 26.1 Six-membered ring 0.8 Cl 55.2 Seven-membered ring 4.0 11 June 29, 2005 Quayle O.R. Chem. Revs. 53 (1953) 439 Example: ethanol H H Unit Contribution to parachor H C C OH O 19.8 C 9.0 H H H 15.5 3 2 1/4 Ps =1×19.8 +6×15.5+×=2 9 130.8(cm /mol)×(erg/cm ) ρ ==0.789g/cm3;M 46g/mol M V ==58.3 cm3/mol ρ 44 ⎛⎞Ps ⎛130.8 ⎞ 4 22 γ =⎜⎟==⎜ ⎟()2.24 =25.3 erg/cm or 25.3mJ/m ⎝⎠V ⎝58.3 ⎠ close to the actual value of 24 mJ/m2 12 June 29, 2005 Example: poly (ethylene oxide) H H Unit Contribution to parachor C C O O 19.8 C 9.0 H H H 15.5 3 2 1/4 Ps =1×19.8 +×4 15.5 +×=2 9 99.8(cm /mol)×(erg/cm ) ρ ==1.12g/cm3;M 44g/mol M V ==cm3/mol ρ 44 ⎛⎞Ps ⎛99.8 ⎞ 4 22 γ =⎜⎟==⎜ ⎟()2.54 =41.6 erg/cm or 41.6mJ/m ⎝⎠V ⎝39.28 ⎠ close to the actual value of 43 mJ/m2 13 June 29, 2005 Contact between solids and liquids Solid/liquid pair wetting partially wetting non-wetting air Change in Gibbs free energy dG when the drop spreads γl liquid an infinitesimal amount dA cos θ θ' θ dG = γ sldA−+γγsdA ldAcosθ γs dA γsl At equilibrium dG /0dA = →−γ sl γγs +l cosθ=0 γl and θ are directly measurable; γ − γ cosθ = s sl γs and γsl are not γ l wetting behavior dependent on interfacial and solid surface energy as well 14 June 29, 2005 2. Surface tension of solid surfaces (γs) Methods for determining the surface tension of solids a) By measuring the contact angle between the solid and liquid b) By determining critical surface tension ( γ cr ) according to Zisman (1964), with the assumption that γ cr ≈ γ s c) By extrapolating surface tension data of polymer melts to room temperature (Roe, 1965; Wu, 1969–71) No direct method available for measurement of surface tension of solid polymers 15 June 29, 2005 2a. Estimation of solid surface energy from contact angle 1/3 1/2 4(VV) γγ=+γ−2(Φγγ) sl sl s l s l Φ= 2 γ sl −+γγs l cosθ=0 1/3 1/3 ()VVsl+ 2 Girifalco and Good (1956) ⎛⎞()1c+ osθ γγ≈ ⎜⎟ sl⎜⎟2 ⎝⎠4Φ M Example: PMMA 3 PMMA 3 ρ ==1.17g/cm ;M 100.1g/mol VPMMA ==86.5 cm /mol PMMA PMMA ρ 68–72o PMMA M ρ ==1g/cm3;M 18g/mol V ==18 cm32/mol; γ =72mJ/m HO22HO HO22ρ HO 2 4(18×86.5)1/3 ⎛⎞ (1c+ os(69)) 2 Φ= 2 =0.93 ⎜⎟ 1/3 1/3 γ s ≈≈72.8 2 38 mJ/m ()18 +86.5 ⎜⎟40× .93 ⎝⎠ By means of the boxed equation, surface tension of solid surface can be calculated from measurements of contact angle and surface tension of liquid16 June 29, 2005 2b.