Polymers: Melts, Crystals, Mixtures and Gels

Transcription

1 7-9-9 Polymers: Melts, Crystls, Mixtures n Gels nres. Dhlin Lecture 5/6 Jones: 5., 6.-6., 7.7, 8.-8., 9. Hmley:.5,.7-.8,.-. hlin@chlmers.se Soft Mtter Physics Outline From the previous lectures we know quite bit bout how polymers behve, especilly in solution n on surfces. We will now look closer t concentrte solutions n interctions between polymer molecules. Coil behvior in polymer melts. Polymer crystls. Polymer mixtures n self ssembly of block copolymers. Polymer gels n the geltion process Soft Mtter Physics

2 concentrtion concentrtion Concentrte Solutions Consier polymer solution with high concentrtion such tht the iniviul coils overlp. This will influence the exclue volume entropy: There is no longer so much free volume vilble to expn into! ilute solution C = position C > highly concentrte solution position Soft Mtter Physics Overlpping We now look t the totl polymer concentrtion in the solution. bove some criticl concentrtion C crit overlpping will occur. Roughly, this shoul hppen when there re more thn monomers in the volume R F. For concentrtion in terms of monomers we cn write: C crit R F = Since R F = /5 we get scling lw: C crit 4/5 /5 Mss-bse concentrtion with monomer weight m gives: C crit m m M /5 / Soft Mtter Physics 4

3 7-9-9 Exercise 5. Poly(imethylsiloxne) is issolve t concentrtion of gl -. The coils re just t the limit of overlpping. The monomer cn be written s C H 6 OSi n hs length of 4 Å. The Kuhn length is. nm t these conitions. Wht is the egree of polymeriztion? Soft Mtter Physics 5 Exercise 5. t the criticl monomer concentrtion we hve: C The numertor is just the mss, but the enomintor nees rescling. The result is: C crit crit m b 9/5 6/5 4/5 We nee the monomer mss: m = = 74 gmol - Solving for n inserting the vlues (with consistent units) gives: m / / 4 m 9/5 6/5 6/5 b C crit 9/ Soft Mtter Physics 6

4 ensity Polymer Melts t high enough concentrtions we cn expect tht the Flory rius no longer works for clculting R! If we increse the polymer concentrtion, eventully we rech sitution with no solvent, polymer melt. In the melt the concentrtion of monomers is expecte to be homogenous. There is no longer ny free volume vilble to expn into. If there re no effects from solvent or exclue volume we re left with configurtionl entropy lone! s result, the verge en to en istnce for coil in the melt cn be escribe by rnom wlk (R = / ) just like in thet solvent! R Soft Mtter Physics 7 Sttes of the Polymer Melt Sttes for polymer melts: Liqui. Crystlline (soli). Glss (soli). Viscoelstic. glss glss lethery crystlline morphous Fully elstic behvior requires cross-links! Wht etermines if we get polymer crystls or glss when cooling? rubbery liqui T g T m temperture (T) Soft Mtter Physics 8 4

5 7-9-9 Crystlliztion Polymers in melt cn crystllize when cool enough. (Crystlliztion cn sometimes lso be inuce by stretching!) The crystlline regions re clle lmelle, in which the chins re ligne long the sme xis. The lmelle thickness is smller thn the contour length, typiclly nm. One coil cn be prt of severl lmelle! The lmelle regions re present in microscle spherulites, in which the lmelle brnches grow from nucletion center. ~ µm ~ nm Wikipei: Crystlliztion of polymers Soft Mtter Physics 9 Semi-Crystllinity Polymers o not become full crystls. Typiclly -8% of the mteril is in crystlline form below T m. Why never %? The chins re entngle in the melt, which mkes it hrer for them to rrnge into lmelle, but on longer timescles it shoul be possible Some fctors influencing egree of crystllinity: Rnom stereochemistry. rnching. Polyispersity. morphous semi-crystlline l However, even polymers tht re essentilly perfect (low PDI, no brnches etc.) still only become semi-crystlline (l < r mx ) Soft Mtter Physics 5

6 free energy free energy Repetition: rrhenius Kinetics The bsic moel of rection kinetics is tht therml fluctutions cn mke system (or prt thereof) rech the ctivte stte, corresponing to the ctivtion energy, fter which the energy chnge is just ownhill. The probbility tht rection occurs is n exponentil function of the ctivtion energy. If the rte constnt is k we hve ccoring to rrhenius kinetics: G k exp kt Most molecules wobble roun with frequencies on the orer of GHz ΔG* ΔG rection progression Soft Mtter Physics The Growth Mechnism Lmelle hve l < r mx ue to kinetics rguments n not equilibrium thermoynmics. ssume tht the polymer must first stretch out istnce l in n (entropic) ctivtion step! ext, (enthlpic) energy is relese when the stem joins the lmelle. Thick lmelle require the formtion of longer stright regions. Thus, the ctivtion energy ( TΔS stretch ) is too high! Thin lmelle o not provie strong riving force since the enthlpic energy relese ( ΔH grow ) is too smll! ΔH grow Tht ws qulittive. Cn we get quntittive moel? -TΔS stretch lmelle growth Soft Mtter Physics 6

7 7-9-9 Entropy of Stretching ssume tht t ech freely jointe chin point there re n iscretize possible irections of which only one represents going stright he. The probbility of going stright is thus p = /n. y combintorics, the number of orienttionl microsttes for stem of length l re: p W bent = [n ] l/ = [/p ] l/ W stright = l/ = The entropy of stretching is thus: l / l Sstretch Since p is most likely smll: S stretch l k log Wstright k log Wbent k log k log p p l k l log k p l log p ote tht since p <, ΔS stretch is negtive s expecte when stretching Soft Mtter Physics l Enthlpy of Lmelle Growth Consier ing one polymer stem of length l n cross section re to lmelle. The mss tht is e is then l ρ, where ρ is the ensity of the polymer. Following the principle of unercooling the enthlpy of crystlliztion per stem is: G grow H mt l T m Here ΔT = T m T is the unercooling (negtive). The melting temperture of the crystl is T m. We efine ΔH m s positive, i.e. the energy neee to melt stems from the lmelle. (It is efine per mss of polymer crystl.) l Soft Mtter Physics 4 7

8 rte Moelling Crystl Growth Using rrhenius kinetics, we cn write the rte t which stems join the crystl s: k join k TS stretch Sstretch exp k exp k exp kt k logp Here we hve introuce chrcteristic frequency k representing the rte t which polymer coils fluctute in shpe. The rte t which stems leve the crystl is: k leve k G exp kt grow k H mtl exp ktmt k This represents jumping bck over the hill to the higher energy stte. The sme chrcteristic frequency k is use. The net growth rte is then: k grow l k join k leve l k exp log p Quite mny prmeters n some re hr to etermine... However, we cn see tht l ppers in two exponentilly ecying terms Soft Mtter Physics 5 l H ml exp k Tm H ml exp k Tm < < T T Mximum Growth Rte Consiering the expression for k grow, there must be mximum growth rte for l * n it will epen on the unercooling ΔT. Thus we hve n explintion why polymers cnnot crystllize fully: The growth rte for lmelle of very high (mx(l) = ) or very low (min(l) = ) pproches zero. k k join (stretching entropy) l kgrow l kgrow k leve (melting enthlpy) l * Soft Mtter Physics 6 lmelle thickness (l) 8

9 7-9-9 Melting Point Wht etermines the melting point n glss trnsition temperture? Extreme vritions epening on polymer type, n chin properties! itives cn be use to chnge T g n T m. University of Cmbrige Soft Mtter Physics 7 Crystllinity Mkes Things Difficult The crystlline regions mkes polymers heterogeneous mterils on the microscle n gives certin properties: The polymer cn be soft mteril in the non-crystlline regions (bove T g ) but lso hr soli in the crystlline regions. Incresing crystllinity cn mke the polymer hrer but lso more brittle. (It is ifficult to brek, but even smll eformtion will brek it.) This hols even bove T g! The polymer becomes opque becuse of light scttering t the mny bounries between crystlline n morphous regions Soft Mtter Physics 8 9

10 7-9-9 Liqui Crystllinity in Polymers Stiff polymer chins cn form liqui crystls, semi-orere strns. This is not the sme thing s lmelle, which re truly crystlline, lthough the regions in between re not. Polymer liqui crystls form the bsis for strong engineering mterils like Kevlr. s with lmelle crystlliztion, the thermoynmics re escribe by the energy of the moleculr interctions n the entropy loss upon liqui crystl formtion Soft Mtter Physics 9 Polymer Mixing Cn we mix polymers (bove T m )? We consier the regulr solution lttice, but the probbility of hving the sme neighbor must increse for polymers! We consier the regulr solution lttice with n sites in totl n one monomer occupying ech site. For one polymer of type, mixing mens going from nφ to n sites so: S W k log W finl initil k log n nφ The totl entropy of mixing cn be written s nδs mix where ΔS mix is the entropy of mixing per site. Further, the number of polymer molecules of type X is nφ X / X. For mixture of n polymers of type n n polymers of type we cn write: ns mix n S n S k Here n cn be remove s expecte! nφ k logφ log nφ Φ logφ Soft Mtter Physics

11 7-9-9 Flory-Huggins Theory We know the efinition of χ is the sme whether we re eling with polymers or solvents. Thus the energy of mixing (per site) for polymer mixture shoul be the sme s in the regulr solution moel: U mix k TΦ Φ ow we cn write the free energy of mixing for binry polymer mixtures: F mix U mix TS mix Φ kt log Φ Φ logφ Φ Φ This moel is known s the Flory-Huggins theory. Snity check: If = = we recover the regulr solution moel! We now hve smller entropy terms. If = = we only hve to replce χ with χ to see tht the criticl vlue for stble mixtures t ll compositions is: Soft Mtter Physics Mixing Polymers is Difficult! The severely reuce entropy gin often mkes it hr to mix ifferent polymers. This cuses problems when trying to recycle plstics! M. Rubinstein, R.H. Colby Polymer Physics Oxfor Soft Mtter Physics

12 7-9-9 lock Copolymers Since ifferent polymers generlly o not mix, block copolymers will ten to self ssemble into microphses. For two blocks one intuitively expects lmelle rrnge perioiclly in D. For long, too much stretching. For short, more unfvorble interfce Soft Mtter Physics Thickness of Microphses The microphses re like melts, so we hve rnom wlk with en to en istnce : G stretch kt constnt umber of chins per volume: /[ ] Interfcil energy per volume: γ/ Interfcil energy contribution per coil is then: γ / So the totl energy per coil is: G tot G constnt tot k T 5 kt / 5 k T / kt Scling reltion is to /. Wht hppens if? block ( monomers) block ( monomers) Soft Mtter Physics 4

13 7-9-9 Other lock Copolymer ssemblies In principle one shoul get two ifferent for the two blocks. However, when is very ifferent from other microphses cn form inste of sheets to reuce interfcil re Soft Mtter Physics 5 Gels: Loose Definition mbiguous use of the term gel is gel until someone proves it is not gel. t lest it seems cler tht gels re soft mtter! Cn contin high volume frction of liqui (which stys boun insie). Superbsorbing gels cn contin 99.9% solvent! Volume chnge is blnce between osmotic pressure n elsticity of the gel mtrix. ot ll gels re bse on polymers! Ono et l. ture Mterils Soft Mtter Physics 6

14 7-9-9 Demonstrtion: Wter bsorbing Gels Cross linke polycrylmie bsorbs lot of wter Soft Mtter Physics 7 Gels: Strict Definition gel is mteril contining some kin of subunits tht re linke together, forming single mcroscopic network. When cross-links re forme between the subunits, we refer to this process s geltion. The mixture goes from being liqui or viscoelstic mteril to soli! Soft Mtter Physics 8 4

15 7-9-9 Gel Types We cn ivie the gels into two types epening on the type of cross links: Chemicl gels: Covlent chemicl bons, usully irreversible. Physicl gels: Other types of interctions, usully reversible by chnging temperture! Jelly Shot Test Kitchen ike SC Soft Mtter Physics 9 Chemicl Geltion of Polymers Chemicl geltion mens formtion of n infinite network by irreversible bons. Vulcnize rubbers. Direct rection cross-links polymers, usully the nturl rubber poly(cis-isoprene). poly(cis-isoprene) Thermosetting resins. polymer with rective en groups n cross linker thn cn bin severl of these groups. The mixture becomes hr n stiff. (Two component glues.) Soft Mtter Physics 5

16 7-9-9 Physicl Geltion of Polymers Recll: Physicl geltion mens formtion of n infinite network by some form of reversible connections. Microcrystlline regions. The lmelle cn ct s cross-links between chins. One exmple is kitchen jelly consisting of the protein mixture geltin (notice the nme) from collgen tissue. The pepties form crystlline omins with helix structures. Microphse seprtion. Use triblock copolymers to get phse seprtion on the microscle. One exmple is polybutiene with polystyrene ens. Cross-linking when T < T g for polystyrene Soft Mtter Physics Moelling Geltion So gels re clerly lots of fun, but wht bout some gel theory stuff? Remember tht gels re chrcterize by hving single network of connecte subunits. s cross-linking bons re forme, there is n brupt chnge in mteril properties when the bons re so ense tht everything connects. From liqui or viscoelstic to soli! How cn we moel this process? ssume we hve subunits which cn form bons with up to z neighbors. (ot to confuse with regulr solution moel.) z = 4 f / The probbility of fining bon, i.e. the frction of recte bining sites is f Soft Mtter Physics 6

17 7-9-9 Percoltion Theory ssume tht we strt with one subunit in n infinite lttice n wlk long one bon to nother subunit. This secon genertion subunit will be connecte to f[z ] new units. The sme hols when we wlk to the thir genertion n so on. If we count the number of bons n connecting to genertion i we get (on verge): i n f z i i = When i we get n or n epening on whether or not f[z ] >. The percoltion threshol is thus: fc z i = 7 It is thus not surprising tht gel forms very suenly uring the geltion process! n(i = 8) = f c = / Soft Mtter Physics The Gel Frction ot ll of the forme bons re prt of the infinite cluster! There will lso be seprte finite networks. The gel frction p gel is efine s the probbility tht bon is prt of the infinite cluster. How cn we etermine it? Let us enote the probbility tht brnch or bining pth from site (not necessrily n ctul bon) is not prt of the infinite cluster s: p The probbility tht none of neighbors sub-brnches (secon genertion bining sites) le to the infinite cluster is then: p z The probbility tht bon ctully exists to given neighbor, but without going to infinity through tht neighbor, is then: fp z Further, the probbility tht bining site is not prt of the infinite cluster (p ) is equl to the probbility tht there is no bon ( f) plus the probbility tht such bon oes not go to infinity (fp z ). Therefore we cn write: p f fp z Soft Mtter Physics 4 7

18 7-9-9 Clculting the Gel Frction The probbility tht site is not connecte to infinity is p z. The probbility tht bon exist without connection to infinity is fp z. Per efinition, this must be equl to the totl frction of recte bining sites (f) multiplie by the probbility of not being in the gel frction: fp z f p gel So we get p gel s: z pgel p ow let us look t ifferent z. For z = no geltion is possible. If z = the solution is trivil since f c = p gel =. For z = things strt to get interesting We first solve for p : p p f f f f fp p p f 4 f f f / 4 f 4 f f 4 f / f / f f 4 f f Soft Mtter Physics 5 Exmple with Three ining Sites The solutions re given by: f p f We thus get the gel frction s: p gel p z f p f Grph shows p gel s function of f for z =. uninteresting gel frction (p gel ) ote tht we will get p gel < when f < f c = /[z ]. The grph ppers t the percoltion threshol frction recte bons (f) Soft Mtter Physics 6 8

19 7-9-9 Exercise 5. brnche polymer hs rective en groups tht cn cross-link with ech other. n brupt chnge in mteril properties is observe fter certin cross-linking time. t this time the probbility tht n en group hs not recte is /. ssuming there re no self cross-links (to the sme molecule), escribe the brnching of the polymer! Soft Mtter Physics 7 Exercise 5. The percoltion threshol is / = / n thus: z 4 f c Thus there re 4 en groups, which mens there must be two brnching points Soft Mtter Physics 8 9

20 7-9-9 Reflections n Questions? Soft Mtter Physics 9 Exercise 5. lmelle grows s circulr isk. The number of fusing stems with volume v per unit time is proportionl to the expose rim re with rte constnt k growth (unit of per re n time ). Wht is the ril growth rte? k growth v Soft Mtter Physics 4

21 growth rte Exercise 5.4 Consier the expression for k grow where the ctivtion step is (entropic) coil stretching n the reverse ctivtion energy is (enthlpic) melting. Show tht the lmelle growth rte is monotoniclly incresing function with respect to unercooling mgnitue. unercooling Soft Mtter Physics 4 Exercise 5.5 Derive the vlues of χ in Flory-Huggins theory for which ll mixtures re stble for two polymers with egree of polymeriztion n. (Do not o the ssumption of = =.) 4 / 4 / Soft Mtter Physics 4

22 Soft Mtter Physics 4 Exercise 5.6 block copolymer contins two regions with egree of polymeriztion n tht hve the sme monomer size n flexibility. The interfcil tension between microphses is γ. ssume sheets re forme n tht their vlues for thickness re = = /. Derive n expression for! Soft Mtter Physics 44 Exercise 5.6 If the monomers re the sme in both blocks the rnom wlk entropy is: The interfcil free energy per volume is n verge weighte with the thickness vlues: The volume of one coil is [ + ] n the totl free energy is: stretch constnt constnt T k T k G T k G 7 constnt tot tot 7 T k G / T k T k