sign in sign up
<<
Home , Solubility, Surface energy, Surface tension, Weight

of Polymers

Abhinandan Agrawal

Hatsopoulos Microfluids Laboratory Department of Mechanical Massachusetts Institute of Technology

1 June 29, 2005 Physical origin of /surface

• “Unhappy” at the surface: they are missing half their attractive interactions

• Unbalanced for the molecules at the surface lead to additional energy

• The additional free energy at the surface is Examples of “minimal ” known as

• This is the fundamental reason behind adjusting their shapes to expose the smallest possible

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

3 June 29, 2005 Estimation of surface tension based on intermolecular forces U/2 U is the energy per inside the

U Energy shortfall for a molecule sitting at the surface ~ U/2 a is the 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 of 25 oC, kT is equal to 1/40 eV, which gives γ = 20 mJ /m2 (close to actual value for and )

Surface tension of a few common liquids in mJ/m2 Water Molten (4 K) (100oC) 0.1 23 24 63 73 58 ~300 485

U ≈1eV 4 June 29, 2005 bonding strongly cohesive liquid Mechanical definition: as a capillary

Surface tension (γ) can also be viewed as a force per unit (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 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 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 of cohesion internal , 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 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 of liquid and vapor at same temperature. C is a a characteristic valid for different compounds constant for a given liquid. and at different

3 2 1/4 Sugden S. J. (Chem. Soc., 1924, 125, 1177) Unit of Ps (cm /mol)×(/cm ) or (m3/mol)×(mJ/m2)1/4 M PC==1/4M γ 1/4 at low temperature, s Dd− d becomes small molecular V

Ps for different substances is a comparison of molecular 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 , V is the , 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 ( )

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 and liquids

Solid/liquid pair

partially wetting non-wetting

air Change in dG when the 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 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

1/3 2 4(18×86.5) ⎛⎞(1c+ os(69)) Φ= =0.93 ⎜⎟2 2 γ s ≈≈72.8 38 mJ/m ()181/3 +86.51/3 ⎜⎟40× .932 ⎝⎠

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. Determining solid surface tension using Zisman plot

Toluene Propylene Zisman found that cosθ is usually a 1.0 monotonic function of γ l 0.8 cosθ =−abγβll=1−(γ−γcr) ) θ

( γ is called the “critical surface tension” 0.6 cr Glycerol of a solid and is a characteristic property cos of any given solid 0.4 Water 2 γ cr = 38 mJ/m Any liquid with γl <γcr will wet the surface 30 40 50 60 70 80 Surface Tension (mN/m) It is found that critical surface tension is close to the solid surface tension of Zisman plot for PMMA using various testing liquids polymer

γ sc≈ γ r

17 June 29, 2005 2c. Estimation of γ from extrapolating data of polymer melt

Extrapolation of surface tensions of melts to room temperature leads to reliable values for solid polymers 4 ⎛⎞Ps Hence, surface tension of solid polymers may be calculated using γ = ⎜⎟ ⎝⎠V Example: PMMA CH3 Ps = 8×+H 5×C +2×O +1×double bond CH2 C

C O Ps = 8×+17.1 5×4.8 +2×20 +1×23.2

OCH3 Unit Contribution = 220.8(cm3/mol)×(erg/cm2 )1/4 to parachor 3 O20 ρ ==1.17g/cm ;M 100.1g/mol C9 44 M ⎛⎞Ps ⎛220.8 ⎞ 4 V ==86.5 cm3/mol γ =⎜⎟==⎜ ⎟()2.55 H17.1 ρ ⎝⎠V ⎝86.5 ⎠ Double 23.2 22 bond =42.5 erg/cm or 42.5mJ/m 18 June 29, 2005 Example: contact angle of water on PMMA

1/2 1/3 4(VVsl) ⎛⎞γ s Φ= cosθ ≈Φ2 ⎜⎟−1 1/3 1/3 2 ()VV+ ⎝⎠γ l sl

4(18×86.5)1/3 ρ ==1g/cm3;M 18g/mol Φ= =0.93 HO22HO 2 ()181/3 +86.51/3 M V ==18 cm32/mol; γ =72mJ/m HO22ρ HO 1/2 ⎛⎞42.5 cosθ ≈×2 0.93⎜⎟−1=0.43 32⎝⎠72 VPMMA ==86.5 cm /mol; γ PMMA 42.5 mJ/m θ = 65o

contact angle of water on PMMA = 68–72 o

19 June 29, 2005 Temperature dependence of surface tension 4 ⎛⎞Ps 1) Since the parachor Ps is independent of the temperature γ = ⎜⎟ ⎝⎠V 4 ⎛⎞ρ()T γγ= (298)⎜⎟ ⎝⎠ρ(298)

2) Guggenheim (1945) relationship 1+r ⎛⎞T --surface tension of a liquid decreases with temperature γγ=−(0)⎜⎟1 ⎝⎠Tcr --surface tension vanishes at the critical point r=2/9

Tcr is the imaginary critical 2/9 dTγγ11 (0) ⎛⎞ temperature of the polymer −= ⎜⎟1− dT 9 Tcr ⎝⎠Tcr

for low values of T/Tcr , dγ/dT will be constant 20 June 29, 2005 Molecular dependence of surface tension

The surface tension of homologous series tends to increase with increasing molecular weight

The bulk properties (e.g. capacity, tensile strength, specific volume etc.) vary linearly with the reciprocal of molecular weight

-2/3 The surface tension, instead, varies linearly with Mn

At infinite molecular weight, surface tension is finite, γ∞

Polymer k γ ke γγ=−e ∞ ∞ 2/3 polyethylene 36 386 Mn 30 373

Polydimethylsiloxane 21 166 ke is a constant polyethylene oxide 43 343

21 June 29, 2005 “Polymer and Adhesion” Wu S. 3. Estimation of interfacial surface energy

The additional free energy at the interface between two condensed phases is known as interfacial energy

4(VV)1/3 1/2 sl Φ= 2 γ sl −+γγs l cosθ=0 γγsl =+s γl −2(Φγsγl ) 1/3 1/3 ()VVsl+

eliminating γs

2 ⎛⎞()1c+ osθ γ =−γθ⎜⎟cos sl l ⎜⎟2 ⎝⎠4Φ

By means of this equation, interfacial tension can be calculated from measurements of contact angle and surface tension of liquid

22 June 29, 2005 Estimation of interfacial tension

Lorentz-Berthelot Mixing Rules A Aab = interaction energy between unlike molecules ab = 1 1/2 AA, = interaction energy between like molecules ()AAaa bb aa bb Girifalco and Good (1956)

∆Fab = free energy of adhesion ∆Faa = free energy of cohesion ∆Fab −=1/2 Φ ()∆∆FFaa bb Φ = interaction parameter

γ adhesion γb ab

∆Fab =−γ ab γγb −a

γa

γa cohesion

∆Faa =−2γ a γa

4(VV)1/3 γγ=+γ−2(Φγγ)1/2 Φ= sl ab a b a b 1/3 1/3 2 ()VVsl+ 23 June 29, 2005 General expression for the interfacial tension

1/2 Equation is only valid for substances γγ=+γ−2(Φγγ) sl s l s l interacting with additive dispersive forces and without hydrogen bonds. (Fowkes, 1964) 1/2 1/2 2 γγsl ≈−()l γs since Φ≈1for most polymers

Fowkes has suggested that total free energy at a surface is the sum of contributions from the different intermolecular forces at the surface.

da d and a refer to the forces and a-scalar forces (the γ =+γγ combined polar interactions: dipole, induction, and hydrogen bonding

dd1/2 dd1/2 aa1/2 γγsl =+s γl −2(γl γs ) γγsl =+s γl −22(γl γs ) −(γl γs ) for apolar liquids/solids Owens and Wendt extended the formulation

24 June 29, 2005 General expression for the interfacial tension

1/2 1/2 221/2 1/2 γγ=−⎡⎤(dd)(γ)+⎡()γa−()γa⎤ 12 ⎣⎦⎢⎥1 2 ⎣⎢1 2 ⎦⎥

Substances 1 and 2 may either be liquids, or solids, or they may be a combination of a solid and a liquid

d a Liquid γ1 γ1 γ1 n- 18.4 18.4 0 25.5 25.5 0 1/2 1/2 221/2 1/2 γγ=−⎡()dd(γ)⎤⎡+(γa)−(γa)⎤ 12 ⎣⎢ 1 2 ⎦⎣⎥⎢1 2 ⎦⎥ glycol 48 33.8 14.2 methylene iodide 50.8 49.5 1.3

1/2 1/2 221/2 1/2 γγ=−⎡()dd()γ⎤⎡+(γa)−(γa)⎤ 12 ⎣⎢ 1 2 ⎦⎣⎥⎢1 2 ⎦⎥ 22 ⎡ dd1/2 1/2 ⎤⎡a1/2 a1/2 ⎤ glycerol 63.4 37.0±4 26 γγ12 =−()1 (γ2 )+(γ1 )−(γ2 ) ⎣⎢ ⎦⎣⎥⎢ ⎦⎥ water 72.8 21.8±0.7 51

25 June 29, 2005 Calculation of force components of surface tension

ad Liquid 1 (e.g. water): γ11,γ

2 • Measure the surface tension of this liquid (γ 1 = 72.3 mJ/m )

• Take another liquid (e.g. cyclohexane) which is immiscible with Liquid 1 ad22 and is apolar (0γγ22==; 25mJ/m;→γ2=25mJ/m)

2 • Measure the interfacial tension by one of the available methods (γ12 = 50.2 mJ/m )

• Solve the following the two equations simultaneously

22 dd1/2 1/2 a1/2 a1/2 γγ=−⎡⎤()()γ +⎡(γ)−(γ)⎤da 12 ⎣⎦⎢⎥1 2 ⎣⎢1 2 ⎦⎥γ11= γγ+ 1

1/2 γγ==50.2 25 +72 −2 25 d γγda==22.7, 72 −22.7 =49.3 HO22,ch ( HO) HO22HO

26 June 29, 2005 Conclusions

• The specific energy of a polymer can be estimated by means of an additive quantity, the parachor

• It may be alternatively calculated from different related material properties e.g. cohesive energy, compressibility, solubility parameter

• Rules are given for the estimation of the interfacial tension and the contact angle of a liquid on a solid

27 June 29, 2005 Extra slides: estimation of Φ

∞∞∞ ∂εab Fzab ()= 2π nbdj fdf na dr ∫∫zj∫f∂r

∞∞ ∞ ∞ ∂εab −∆Fab ()z = 2π dx nbd j f df na dr ∫∫zx ∫j ∫f∂r

AB Carrying out the integrations and equating the net force to 0 Lennnard-Jones ε =− ab + ab ab rr6 m when z=dAB (equilibrium distance) potential ∂ε 6A mB Interaction ab =−ab ab nnabAab⎛⎞11 71m+ ∆=Fab 2 ⎜⎟− force ∂rr r 62dmab ⎝⎠− 4 2 nAaaa⎛⎞11 ∆=Faa 2 ⎜⎟− 62dmaa ⎝⎠− 4 1/3 ∆F dd A 4(VVsl) −=ab ΦΦ= aa bb × ab Φ= 1/2 2 2 1/3 1/3 2 ∆∆FF dab ()AA ()aa bb aa bb ()VVsl+

28 June 29, 2005 Girifalco and Good (1956) 1) Floating needles l

(F1+F2)cosθ

θ F F2 1 W end view of needle 29 June 29, 2005 F1=F2=γl