Rheology of Dilute Polymer Solutions with Time-Dependent Screening of Hydrodynamic Interactions

Transcription

1 arxiv:ysics/cond-ma.sof Reology of Dilue Polymer Soluions wi Time-Deenden Screening of Hydrodynamic Ineracions V. Lisy, J. Toova,. ruovsky Te screening of ydrodynamic ineracions (HI) essenially affecs macroscoic roeries of olymeric soluions. Tis screening deends no only on e olymer concenraion, bu as a dynamic naure. In e resen work, a bead-sring eory is develoed, in wic is enomenon is described for soluions of nonenangled olymer coils. Te equaion of moion for e beads of a es olymer is solved ogeer wi e rinkman s equaion for e solven velociy a akes ino accoun e resence of oer coils in soluion. Te ime correlaion funcions for e olymer normal modes are found. A endency o e screening of HI is demonsraed on e coil diffusion as well as on e relaxaion of is inernal modes. Wi e growing concenraion of e coils ey bo sow a ransiion o e exac Rouse beavior. Te viscosiy of e soluion and some oer observable quaniies are calculaed. As e ime increases, e ime-deenden quaniies cange eir beavior from e Rouse regime roug e imm one again o e Rouse dynamics a long imes. I. Inroducion Reology of olymer soluions reresens a crossdiscilinary field, using wide secra of eoreical ools from ysics and cemisry. For ysiciss, undersanding e configuraion and dynamics of long olymer cains as been a significan source of roblems wiin saisical ysics from e 95 s onwards. One of e reasons wy ysiciss were drawn o e roblem is e universaliy of olymer roeries []. Wiin e ime and leng scales muc exceeding e aomic ones, universal eories ave been buil, well describing e main feaures in e olymer beavior, insensiive o e deails of e cemisry of e cains. Among ese eories e mos oular are e Rouse and imm models, in wic e olymer is reresened as a cain of beads under rownian moion []. Te resen work was insired by e difficulies a sill exis beween ese models and exerimens. For examle, one can find differen resuls for e viscosiy of dilue olymer soluions (see [-4] and e ciaions ere), e observed monomer moion in single olymer cains canno be exlained by e available eories [5, 6], ere are roblems wi e descriion of e dynamic lig scaering from olymer soluions [7, 8], and oers [9]. Aloug e Rouse and imm models give a good qualiaive base for e olymer soluion reology, some of e exising quaniaive discreancies wi e exerimens are no resolved for decades and new roblems aear. An examle is e comuer simulaion sudy [], were e screening of ydrodynamic ineracions (HI) in semidilue olymer soluions was invesigaed. I is well known a, due o is screening, wi e increase of eir concenraion e olymers in soluion cange eir beavior from e imm- o e Rouse-like one []. I as been found in [] a is rocess is no only concenraion deenden bu as a dynamic caracer, i.e., i canges wi e ime. ased on our revious generalizaion of e Rouse-imm eory [, ], is observaion can be exlained eoreically in a naural way. Our aroac o e roblem can be summarized as follows. Firs, as disinc from e radiional use of e imm model for olymers in soluion (is corresonds o srong HI and is in e eory caracerized by e infinie draining arameer []), we consider is model jus as a secial case of a more general eory wi a finie. Due o is, bo e diffusion coefficien of e wole olymer coil and e relaxaion raes of is inernal modes are sums of Rouse and imm conribuions, e former one being usually omied in e inerreaion of exerimenal daa. Secondly, e inernal modes ave eir own draining arameer (), wic deends on e mode number,,... Wen increases, () decreases and beginning from some all e iger inernal modes become e Rouse modes even if e wole olymer is redominanly of e imm ye. Tird, e ye of e dynamics deends also on e ime. Consider, for examle, e mean square dislacemen (MSD) of a olymer segmen. A sufficienly sor imes is dynamics in a flexible olymer is Rouse-like and wi e growing ime i canges o e imm-ye dynamics (e relaive conribuion of e modes wi small draining arameers decreases, or, in oer words, e influence of e HI increases) [6]. Tis crossover canno be described coming from aroximae exressions, according o wic e MSD is given by e simle / or / laws for e Rouse or imm olymers, resecively [, 4]. Tese laws ave been obained assuming e coninuous disribuion of inernal modes wi resec o. However, is assumion is rue only

2 V. Lisy, J. Toova,. ruovsky in a resriced ime domain and can lead o incorrec inerreaion of exerimenal daa [5]. Using e discussed aroac, we ave recenly [5, 6] inerreed e fluorescence correlaion secroscoy measuremens on individual DNA molecules and sowed a e original inerreaion [5] of e daa was misleading. Wen e moion of a macromolecule is affeced by oer coils in e soluion, one more roblem arises: i is necessary o ake ino accoun e forces wi wic e coils ac on e solven and us amer is flow. For nonenangled olymers i can be done using e Debye- uece-rinkman eory for e flow beween obsacles [6, 7]. Te fricion facor f on e olymer cain during is moion can be deermined from e Einsein relaion D k T/f, were D is e coil diffusion coefficien. y is way, in e nex secion e generalized Rouse-imm equaion for e osiion vecors of e beads and e Oseen ensor describing e velociy field of e solven due o erurbaions are buil. Te described aroac en allows us o obain new resuls on e diffusion of e olymer coils and eir inernal moion and o find e relaed observable quaniies, suc as e viscosiy funcions of e olymer soluion or e relaxaion modules. Wi e growing concenraion e imm conribuion o ese quaniies disaears. Our eory us describes in a simle manner e endency o e ransiion beween e imm and exac Rouse beavior of e olymer. Along wi e concenraion deendence, e dynamic naure of e screening of HI is revealed: as e ime increases, e ime-deenden quaniies, suc as e bead MSD, cange eir beavior from e Rouse regime roug e imm one again o e Rouse dynamics. II. HYDRODYNAMICS OF POLYMER SOLUTIONS We coose one of e olymers in soluion as a es olymer []. Te equaion of moion of is n bead is d xn ( ) fr c M f n + fn + fn. () d Here, x is e osiion vecor of e n bead from N c ones maing e olymer, M is e bead mass, f n e force wi wic e neigboring beads ac on e n bead, f is e random force due o e moion of e n fr molecules of solven, and f n is e Sokes fricion force on e bead during is moion in e solven [8], fr dxn fn ξ v ( xn ) d, () were v ( x n ) denoes e velociy of e solven in e lace of e n bead due o e moion of oer beads. Te fricion coefficien is ξ 6πηb (b is e bead radius and η e solven viscosiy). Tis exression olds in e case of seady flow. In a more general case aking ino accoun e ydrodynamic memory [, 9, ] e force () mus be relaced by e oussinesq force and () as o be solved ogeer wi e nonsaionary ydrodynamic equaions for e macroscoic velociy of e solven. To ake ino accoun e resence of oer olymers in soluion, we use e rinkman-debye- uece [5, 6]) eory in wic e olymer is considered as a orous medium. In our aroac all e soluion is suc a medium ermeable o e solven flow. Ten in e rig and side of e Navier-Sokes equaion a erm -κ η v as o be added, were /κ is e solven ermeabiliy. Tis erm corresonds o e average value of e force acing on e liquid in an elemen of volume dv, rovided e average number of olymers in soluion er dv is c; en κ η cf, were f is e fricion facor on one olymer cain. Tus, ρ v v v η + κ η + ϕ. () Here, ρ is e solven densiy, is e ressure, and ϕ is e densiy of e force from e olymer beads on e liquid [8], ϕ ( x) f fr n ( xn ) δ ( x xn ). (4) n Solving is equaion is a difficul roblem since e olymer coils are moving. However, in e firs aroximaion, small and slow canges of e concenraion c() around is equilibrium value can be negleced. Te beads are muc more mobile an e wole coils of long olymer cains (N >> ). Tis is seen comaring e diffusion coefficien of one bead, D b k T/(6πbη), wi a of e coil in e imm (D 8k T/[(6π N) / aη]) or Rouse (D R k T/(6πNbη)) limis [] (a is e mean square disance beween e beads along e cain). In e laer case D R /D b /N, and for e imm olymers D /D b.7b/(a N). Te moion of e solven creaed by e moion of beads is us muc faser an e moion of e coils, wic deermines e canges of c(). Eqs. () - (4) describe e moion of one bead in e solven, wen e obsacles (oer coils) influence e solven flow. Tis roblem can be ransformed o a firs solved in [] (for laer works see also [, ]). Te velociy field can be in e Fourier reresenaion in e ime wrien as follows: v r dr H r r r. (5) ( ) ( ) ϕ ( ) α αβ β β Here, e analog of e Oseen ensor is were r r α β H r A, (6) r αβ ( ) δαβ + y y e A e y, 8πηr y

3 V. Lisy, J. Toova,. ruovsky y y e e + y, (7) 8πηr y y rχ, χ κ - iρ/η, and e rime means e differeniaion wi resec o y. In e aricular case and for ermeable solvens wen κ, Eqs. (6) and (7) coincide wi e resul of imm []. Using is soluion, a generalizaion of e Rouse-imm equaion as been obained from e equaion of moion []. Te reaveraging of e Oseen ensor over e equilibrium (Gaussian) disribuion of e beads [, 8] gives wi H δ ( n m), r nm x n x, (8) m αβ nm αβ / ( ) ( 6π ) ( η ) n m n m a ( ) ( ) π z ex z erfc z, z χa n m / 6. Ten, in e coninuum aroximaion wi resec o e variable n [, 8], e new Rouse-imm equaion conains only e diagonal erms [], kt x ( n) i x ( n) + M x ( n) + f ( n) ξ a n N k x m T + dm ( n m) a m ( ) + M x ( m) + f ( m). (9) I is solved wi e el of e Fourier ransformaion (FT) in n, ( ) x n y + y cos ( π n / N ), aking ino accoun e condiions a e ends of e cain, x, n / n a n, N. Te inverse FT yields ( ) were y f iξ M + K ( ) Ξ ξ + δ Nξ, () π, K kt Na,,,, and e Oseen marix reads [], + ex erfc 6Nπηaχ π χ χ + χ πη a π N + + χ ( ) () ( χ χ ),,,,... () were χ N / 6χa and χ N /( π ) χa. Using e flucuaion-dissiaion eorem [4] or e roeries of e random forces [5], e ime correlaion f funcions for e normal modes are deermined as ( ) y ( ) y ( ) ψ () α α k Re T Ξ δ π d cos. N i Ξ M + K ( ) In e saionary limi so a χ κ. Ten e reaveraged Oseen ensor (6) is δ αβ H αβ 6πη χr e r, (4) and e quaniy /κ can be for small κr considered as a screening leng. Le us firs focus on e moion of e cener of ineria of e olymer. II.. Diffusion of e Polymer Coil For an individual olymer and in Eq. (), ( ) ψ ( ) ψ D, (5) wi e diffusion coefficien D D R + D (R and say for e Rouse and imm limis given above). Now insead of Eq. () we ave wi χ κ R G (R G Na / 6 is e gyraion radius), D deends on e concenraion of e coils c, D( c ) D ( c) + D, D ( ) R D, (6) and consiss of e Rouse (indeenden on e resence of oer olymers) and e imm conribuions. Te laer one can be exressed in e form ( ) ( ) ( ) D c D f c, (7) were f(c) is a universal funcion for every olymer: π ( ) ( ex χ erfcχ ) f c 4χ π χ χ. (8) Te deendence of e ermeabiliy on c can be esimaed as follows. Te fricion coefficien in e quaniy κ cf/η from Eq. () can be deermined from e Einsein relaion D k T/f. Ten κ 7 π c + 6 RG 4. (9) Te values of κ and χ deend on e draining arameer N / π b / a (if >>, e coil dynamics is of e

4 V. Lisy, J. Toova,. ruovsky imm ye, for << we ave e free-draining Rouse limi). Te quaniy c 4 π RG c / denoes e number of olymers er e volume of a sere wi e radius R G. Wi e increase of c e imm erm decreases and for large c (small ermeabiliy κ, wenχ >> ) i becomes ~ / c, D k T. () πη Na κ ( c) Te realisic case of small c corresonds oχ κr G << wen D ( c) kt ( c) D ( ) κ RG +....() 8 π Te c-deenden correcion o D () is roorional o c and differs from oer resuls (e.g. [6], were is correcion is ~ c). Wen e olymer is free, e ye of is diffusion deends only on e draining arameer. Wi e growing c, e olymer canges is beavior o e diffusion wi e exacly Rouse coefficien D R. II.. Dynamics of Inernal Modes In e saionary case and a κ e diagonal elemens of e Oseen marix are [] Now ( ) / ( ) ( π ) ( η ) N a. () c from () deends on c. Te inernal modes relax exonenially, ψ k T () NK ( ) ex ( / τ ) and eir relaxaion raes consis of e Rouse conribuion and e c-deenden imm ar, ( c) ( c) +, (4) τ τ τ R were τ R and τ () τ are given by [] and τ R N a bη, π k T τ τ ( c) τ ( ) wic a c beaves as ( N a) / ( π ) η, (5) k T ( χ ) χ, (6) N τ ( c) τ ( ) + κ a..., (7) 6π and as c one as ( Na ) η τ ( c) τ ( ) χ κ. (8) 6π k T II.. Seady Sae Viscosiy Viscosiy is e mos imoran roery a deermines e flow caracerisics of e fluid. Using e above calculaed relaxaion imes τ of e olymer inernal modes, e seady sae viscosiy of e soluion can be calculaed from e formula [9] η η + τ. (9) ( c) k Tc ( c) In e Rouse limi we obain e known resul [] η(c) η + πn a bcη/6. In e imm case a low concenraions ( c) / 6 ( Na ) c( Na ) η η c / π / η π 6 c( Na ) c( Na ) / / , () were e firs erm corresonds o e known formula []). In our eory, e mos general exression for e viscosiy is η ( c ) η N a b + χ ηc π + () + ( + χ ) A very low concenraions wen χ <<, we ave for e so called inrinsic viscosiy [ η] η η ( c) η η N a b +. () ηc π Fig.. Viscosiy normalized o is Rouse limi as a funcion of <, wen e olymer is assumed o be e Rouse one. Due o e deendence on e difference beween [η] and e classical resuls can be noable. For a Rouse olymer wi small is is illusraed by Fig.. In Fig., [η] >> ~ / is e inrinsic viscosiy of e imm olymer, for wic [η(c) - η] /η.6n a bc/(π). I is 4

5 V. Lisy, J. Toova,. ruovsky seen a even for e difference from e imm viscosiy is ~ %. η η >> Fig.. Te same as in Fig. for large (e imm olymer). Using e above resuls, e Huggins coefficien k H [], wic is one of e mos ofen deermined quaniies in viscosimery measuremens, can be found. From e general exression for e viscosiy (), e Huggins equaion is were k H k H/k Himm.5 η ( c) ηc η + +, () [ η] ( kh [ η] c...) 4 π + +. (4) 7 / + Fig.. Huggins coefficien normalized o is imm limi. In Fig., e Huggins coefficien relaed o is imm limi is sown. I is seen a wi e growing, k H slowly aroaces k Himm. Te difference is significan in a broad region of. For large (e imm case) we find [ η] / a / RGζ 6.5 G π π N were ζ is e Riemann zea funcion. In is case 5 5 R, (5) ( ) ( ) 5/ k π ζ 5 / ζ /.75. (6) Himm Noe a in our work [] e facor / is missing in e exression for k Himm. Tis resul differs from e known resuls (e.g., Doi and Edwards [] give e value.757, k Himm.6949 in [4], ec.; see e discussion in [, 4]). Te works [7, 8] ossess e viscosiy, wic is inconsisen wi e Kirkwood and Riseman [9] eory and gives e ydrodynamic screening even for infiniely dilue soluions. Te work [8] suggess a e screening canno be described if e reaveraging aroximaion is emloyed for e HI; as sown ere, is is no rue. Finally, in e oosie Rouse limi wen, k H aroaces zero as k H πζ(.5)ζ - ().. II.4. Te Relaxaion Modulus Te relaxaion modulus G deermines e sear sress a υ r, ζ r, sear flows wi e velociy ( ) ( ) υ υ [, 8], y z G GR ( ) ( ) + ( ) ( ) σ xy ηζ d G ζ. (7) d G ( ) ζ ( ) Fig. 4. Relaxaion modulus G as a funcion of a. G GR Fig. 5. Te same as in Fig. 4. a.. x y /τ R /τ R Having solved e cain dynamics, e modulus G is calculaed from e equaion 5

6 V. Lisy, J. Toova,. ruovsky G k Tc τ. (8) ( ) ex( / q ) q Figs. 4 and 5 sow G a c, relaed o e Rouse model. Wi e growing, e difference from e Rouse resul becomes significan even a small. So, wen /τ R, even for / e difference beween G and is Rouse limi is abou %. Te numerical calculaions using is exression are given in Fig. 8. We relae e Rouse MSD (a ) o e Rouse-imm MSD a, o sow ow is funcion canges deending on e ime a a relaively low concenraion (one coil in e volume 4π R G /). I is seen a a long imes e beavior of e olymer, wic was iniially redominanly of e imm ye, canges o e Rouse-like ye. G G /τ R 4 5 /τ Fig. 6. Sor-ime beavior of e relaxaion modulus relaed o is imm limi. Te draining arameer is. Fig. 6 illusraes e difference of G () from e ure imm beavior. A very sor imes is difference is significan even for large. Wi e increase of, G becomes closer o is imm limi; owever, a ransiion o e Rouse beavior a long imes is observed, as sown in Fig. 7. For e cosen, e difference from e imm modulus is always larger an %. G G Fig. 7. Te same as in Fig. 6 a longer imes. II.5. Monomer MSD /τ Similar resuls can be obained for oer quaniies, like e comlex modulus, e dynamic srucure facor of e es olymer [8], or e MSD of a monomer wiin an isolaed coil [5, 6]. Te inernal modes of e olymer conribue o e MSD of is end monomer as follows [6]: r ( ) in Na ex. (9) π τ ( c) 6.4 r ( ) r ( ) in, R in Fig. 8. Relaion of e Rouse MSD (a ) o e Rouse-imm MSD a and c. as a funcion of /τ R. Some more calculaions are given in [6] for a single olymer coil and in [8], were e influence of oer coils is considered. Tese resuls can be summarized as follows: every olymer a very sor imes a any c beaves as e Rouse one since e HI does no ye affec e dynamics. A longer imes, e HI akes effec and e olymer begins o move in e imm regime. Ten, due o e screening of HIs, e olymer beavior urns again o e Rouse-like one. III. Conclusion Te roeries of comlex olymeric sysems canno be comreended wiou undersanding e dynamics of a single olymer in well defined condiions, suc as in dilue soluions of nonenangled olymer coils wen, a e scales muc exceeding e aomic ones, only e ydrodynamic forces deermine e olymer beavior. Even is seemingly simle siuaion is no fully described in e lieraure. For examle, e eory of e screening of HI due o e resence of oer coils in e soluion sould be develoed. Te work [] suggess a is screening is no only concenraion deenden bu is a ime-deenden rocess. Te aim of e resen work was o give a descriion of is enomenon and o find ow i reveals in e observable quaniies. As in e radiional eories, we sared wi e equaions of moion for e es olymer, wic sould be solved ogeer wi e ydrodynamic equaions for e solven. Te resened eory as e following limiaions. Te considered imes are >> R ρ/η, were R is e ydrodynamic radius of e coil. Tis means a e ydrodynamic memory effecs [, ] (so far no observed in e olymer dynamics) are

7 V. Lisy, J. Toova,. ruovsky negleced. We are also resriced o θ solvens []; oer cases require knowledge of e equilibrium disribuion of e beads wi e excluded volume ineracions aken ino accoun. Since only soluions of nonenangled olymers are considered, e sudied concenraions of e cains are c < /[η] [9]. Our aroac differs from e revious bead-sring eories in several oins. Firs, we do no a riori assume e validiy of a concree, Rouse or imm, model. Only e sreng of e HI deermines wic ye of e olymer beavior is dominan. Secondly, as disinc from e usual aroximaion leading o simle universal equaions suc as e / (Rouse) or / (imm) laws for e MSD of e olymer segmens [], e disribuion of e inernal modes of e olymer is no coninuous. Wiin e usual aroac i is no ossible o describe e ransiions beween e Rouse and imm beavior of e olymer. However, is ransiion always akes lace since a sor imes e HIs do no affec e olymer moion. Te olymer moves according o e Rouse eory and a longer imes, wen e HIs develo, e regime of is dynamics canges o e imm one. Te conce of e join Rouse-imm model is essenial also for e descriion of e olymer dynamics affeced by oer coils in e soluion. uilding e ydrodynamics of e soluion of nonenangled olymers, we ave sown a wi e increase of e olymer concenraion e imm conribuion o e observable quaniies (suc as e coil diffusion coefficien or e viscosiy of e soluion) decreases and e olymer ends o beave (as disinc from e revious eories []) exacly in corresondence wi e Rouse model. Te same endency is seen wi e increase of ime. Tus, e eory is able o exlain e dynamic naure of e screening of HI. To ake ino accoun e resence of oer coils in e soluion, we ave used e rinkman s ydrodynamics for orous media, adoing i for e solven flow in e soluion were e obsacles o e flow are e olymer coils emselves. Te main resuls of e resened aroac consis in new equaions for e osiion vecor of e olymer beads and for a number of caracerisics describing e beavior of flexible olymers in dilue soluions. Tese quaniies could be verified in sandard exerimens suc as e viscosimery or lig scaering, and in comuer simulaion sudies similar o ose in []. References [] P.-G. de Gennes, Scaling Conces in Polymer Pysics (Cornell UP, Iaca, 979). [] M. Doi, S. F. Edwards, Te Teory of Polymer Dynamics (Clarendon, Oxford, 986). [] M. Muukumar, J. Pys. A: Ma. Gen. 4 (98) 9. [4] M. Muukumar, J. Cem. Pys. 79 (98) 448. [5] R. Suserman, S. Alon, T. Gavrinyov, O. Kricevsky, Pys. Rev. Le. 9 (4) 48. [6] J. Toova,. ruovsky, V. Lisy, Eur. Pys. J. E 4 (7) 6. [7] N. Sawaari, T. Yosizaki, and H. Yamakawa, Macromolecules (998) 48. [8] J. Toova,. ruovsky, V. Lisy, Laser Pys. 7 (7) 44. [9] R. G. Larson, Te reology of dilue soluions of flexible olymers: Progress and roblems, J. Reol. 49 (5). [] P. Alrics, R. Everaers,. Dünweg, Pys. Rev. E 64 () 45 (R). [] V. Lisy, J. Toova, A. V. aovsky, J. Cem. Pys. (4) 699. [] V. Lisy, J. Toova, A. aovsky, Cond. Ma. Pys. 9 (6) 95; arxiv:cond-ma/5998. [] P.-G. de Gennes, Pysics (967) 7. [4] E. Dubois-Violee, P-G. de Gennes, Pysics (967) 8. [5] J. Toova,. ruovsky, V. Lisy, Czec. J. Pys. 55 (5). [6] H. rinkman, Al. Sci. Res. A (947) 7. [7] P. Debye, A. M. uece, J. Cem. Pys. 6 (948) 57. [8] A. Yu. Grosberg, A. R. Koklov, Saisical Pysics of Macromolecules (Nauka, Moscow, 989). [9] J. Toova, V. Lisy, A. V. aovsky, J. Cem. Pys. 9 () 5. [] A. V. aovsky, V. Lisy, J. Mol. Liq. 5 () 89. [] A. V. aovsky, M. V. Levin, V. Lisy, Fizika aerodisersnyk sisem 8 () 8 (in Russian). [] T. Yu. Tcesskaya, J. Mol. Liq. (5) 4. [] T. Yu. Tcesskaya, J. Al. Crysallogray 4 (7) s69. [4] L. D. Landau, E. M. Lifsiz, Saisical Pysics (Nauka, Moscow, 976). [5] P. P. J. M. Scram, I. P. Yakimenko, Pysica A 6 (998) 7. [6] H. ao, W. eckam, H. L. Ricks, U. H. F. unz, Polymer 46 (5) 489. [7] K. F. Freed, S. F. Edwards, J. Cem. Pys. 6 (974) 66. [8] K. F. Freed, S. F. Edwards, J. Cem. Pys. 6 (975) 4. [9] J. G. Kirkwood, J. Riseman, J. Cem. Pys. 6 (948) 565. Auors informaion Dearmen of Pysics, Tecnical Universiy of Kosice, Park Komenskeo, 4 Kosice, Slovakia, Insiue of Pysics, Faculy of Science, P. J. Safarik Universiy, Jesenna 5, 4 54 Kosice, Slovakia Acknowledgemens We dedicae is aer o our unimely deceased colleague, Professor Alexandr Vsevolodovic aovsky. Tis work was suored by e grans //6 and /4/7 from VEGA, Slovak Reublic. 7