Yunshen Cai and Michael Gevelber Department of Mechanical Engineering Boston University, Boston

Transcription

1 Analyss of Electrospnnng Bendng Regon Physcs n Determnng Fber Dameter: focus on mass transfer and effect of relatve humdty for non-aqueous hydrophlc solutons Yunshen Ca and Mchael Gevelber Department of Mechancal Engneerng Boston Unversty, Boston Abstract: Ths paper analyzes the bendng regon of non-aqueous hydrophlc PVP/alcohol solutons, usng a combned expermental and modelng approach. A major focus of ths work s modelng the complex mass transport ncludng solvent evaporaton and water absorpton, whch s verfed by evaporaton experments. The developed model captures the coupled mass and force balances n the bendng regon and predcts the fnal fber dameter to wthn 8% of expermental measurements for three dfferent PVP/alcohol solutons wth sgnfcantly dfferent propertes. The model analyss reveals the effect that RH has on alcohol evaporaton rate, whch affects both net stretchng force and jet length n determnng the fnal fber dameter. 1. Introducton Ths paper studes the role of the electrospnnng bendng regon n determnng fber dameter for non-aqueous hydrophlc PVP/alcohol solutons. The electrospnnng process frst typcally forms a straght jet from a charged polymer/solvent soluton, followed by a crcular movng jet n the shape of a cone, called the bendng regon. The process physcs n the bendng regon s challengng to study snce the jet dameter cannot be measured drectly due to ts rapd moton and small sze (~mcron and smaller), and due to the complex couplng between the multple forces, mass transport, and changng jet geometry. For hydrophlc polymer solvent systems, there s an addtonal complexty snce the solvent absorbs water, and expermentally, t s well known that ambent relatve humdty (RH) sgnfcantly affects the process. To develop a more explct understandng of the process physcs, a bendng regon model s developed whch predcts the jet behavor and 1

2 fber dameter for three solutons (PVP/methanol, PVP/ethanol and PVP/1-butanol) for a varety of process condtons, ncludng RH. The model s valdated by experments conducted for those three solutons, whch have sgnfcantly dfferent propertes, over a broad range of operatng condtons. Analyss of the model results provdes nsght nto how the net evaporaton rate affects the bendng regon jet length and net stretchng force, both of whch ultmately determnes the fber dameter, as well as the mpact of RH. There have been a number of electrospnnng process models developed that relate forces to fber dameters. Hohman, et al. [1, 2] developed a slender-body model for both straght jet and bendng regons that captures jet stretchng, charge transport, and the electrc feld. Ther model ncludes three competng factors: extensonal vscous stress, surface tenson, and electrostatcs. In the bendng regon, they consdered the change of force and electrc feld due to the jet moton, but they do not consder the mass transfer/ evaporaton, nor the mpact of RH. For the straght jet regon, Feng [3] smplfed the slender body model for Newtonan jets usng an approxmaton for the electrc feld equaton, and predcted jet behavor. Hs dfferental format of the steady-state momentum equaton provdes the bass to analyze force balance n the bendng regon n ths research. Frdrkh, et al. [4] analyzed Hohman s dynamc equatons for the whppng jet to obtan an asymptotc soluton that relates the fnal fber dameter to operatng condtons, based on a balance between normal stresses due to surface tenson and surface charge repulson yeldng: d fber ~ ( I Q ) 23. The analyss does not nclude a vscous term, snce t s argued that n the termnal state of electrospnnng process, the vscous force s relatvely small and does not determne fber dameter. They also dd not explctly consder mass transfer effects due to evaporaton rate, although t s expected to determne jet length. It s well known expermentally that RH can have a sgnfcant mpact on the electrospnnng process and resultng fber dameters. For some hydrophlc polymers n hydrophlc solvent systems, the electrospun fber dameters decrease as RH ncreases [5-7]. De Vreze, et al. [5] expermentally found that fber dameters of PVP (hydrophlc) n alcohol (hydrophlc) solutons decrease wth ncreasng RH. They state that the fber dameter decreases at hgher humdty snce the absorpton of water ncreases the tme of flght of the polymer soluton jet. However, they do not analyze the amount of water absorbed under dfferent RH condtons, nor the effect of water absorpton on jet stretchng. 2

3 Ca et al. [6] expermentally found that fber dameters of PVP/alcohol solutons decrease wth ncreasng RH. They present a fber dameter correlaton for PVP/alcohol solutons based on measurable process parameters (upper jet dameter) and RH. However, they dd not determne how RH affects the process physcs that determnes fnal fber dameters. Wang [7] electrospun polystyrene (PS) n THF and DMF solvents to nvestgate the nfluence of soluton and process parameters on the jet dameter (d fnal S,jet ), measured at the end of straght jet regon, and fber dameter (d fber ). They found that d fber ~m d 0.45 S,jet, where m~η K 0.12, n terms of η 0 the shear vscosty, K the conductvty, and soluton vscosty. However, they dd not explctly analyze how soluton and process parameters affect the force balance, nor the effect of evaporaton and RH. To examne the effect of RH, a model of mass transport n the bendng regon s needed. The major challenge for PVP/alcohol solutons s to model mass transfer whch ncludes absorpton/evaporaton of water and alcohol, radal dffuson of both speces n the jet, and axal advecton. Yarn, et al. [8] proposed a model for solvent evaporaton and soldfcaton n an electrospnnng jet by relatng solvent evaporaton coeffcent to Sherwood number. They consdered the change of momentum and vscosty due to evaporaton. However, they dd not explctly analyze the effect of evaporaton and RH on charge densty and stran rate relatve to electrc and vscous forces, nor explctly analyze a second speces (e.g. water) absorbed by the solvent. Forward et al. [9] provdes a mass transport model for the soldfcaton of a co-axal jet, where the core solvent (water) dffuses through the shell solvent and both solvents evaporate at the outermost surface. Although they dd not analyze water absorpton by a solvent, ther work on the water dffuson and solvent evaporaton provdes a bass for our water absorpton and solvent evaporaton model development. 2. Expermental approach Electrospnnng experments are conducted wth 12wt% PVP/methanol, PVP/ethanol and PVP/1-butanol solutons n a system wth controlled RH. The electrospnnng system ncludes the capablty of montorng multple process states n real-tme and settng actuator set ponts for the syrnge pump (Harvard Apparatus, PHD 2000), hgh voltage 3

4 power supply (HV POWER SUPPLY, XRM30P) [10]. A salt bath (magnesum chlorde) system and a water vapor system [11] are used to adjust the expermental RH level. Magnesum chlorde powders absorb water vapor from the ambent ar to decrease the ambent RH. Water vapor generated by a water heatng system s used to ncrease the ambent RH. Two CCD cameras are used to measure the jet dameters n straght jet regon. Jet current s measured by a current sensor, whch s placed between the fber collector and NI USB-6008 data acquston (DAQ) card [10,11,12]. The ambent RH and temperature are montored by HTM 2500 n real-tme. Acheved electrospun fber dameters are maged usng a Zess SUPRA 40VP FESEM (Feld Emsson Scannng Electron Mcroscope). 2.1 Materals PVP wth 1,300k molecular weght were dssolved n dfferent alcohol (methanol, ethanol and 1-butanol) solvents to make 12wt% PVP/alcohol solutons. PVP powders and all alcohol solvents were purchased from the Aldrch Chemcal Co. All solutons were stored n a refrgerator at 5. Before experments, solutons were put n the ambent envronment for several hours to brng the solutons to room temperature (21 24 ). All electrospnnng experments are also conducted at ambent room temperature. Shear vscosty Water absorpton rate PVP/1-butanol Solvent evaporaton rate PVP/ethanol PVP/methanol Conductvty Fg. 1 Normalzed characterstc parameters of PVP/alcohol solutons, n terms of shear vscosty, conductvty, solvent evaporaton rate and water absorpton rate. 4

5 Electrc feld (kv/m) Parameters of the PVP/alcohol solutons are shown n fg. 1, n terms of shear vscosty, conductvty, solvent evaporaton rate and water absorpton rate, normalzed by the maxmum value. The shear vscosty measurements were performed n a coneplate vscometer (TA Instruments AR2000). The conductvtes are measured by a conductvty sensor (TDTestr 40). The alcohol evaporaton rates are publshed n a handbook of organc solvent propertes [13], whch are also verfed by alcohol evaporaton experments under 10% RH (descrbed n appendx. B). The water absorpton rates are estmated based on water dffusvtes n alcohol (see secton 3). Fg. 2 Measurable process parameters 2.2 Expermental results and mplcatons Fg. 2 shows the measurable parameters of the system. The nteracton of the electrostatc forces and the surface tenson of the lqud create a Taylor cone at the end of the needle, whch can be measured wth a characterzed Volume ( ) [10]. When the electrcal force s larger than the surface tenson, a jet emerges from the Taylor cone and s accelerated due to Coulomb forces [10]. The straght jet s characterzed by the jet Flow rate (ml/mn) Fg. 3 Electrc feld operatng bounds: upper (full lne) and lower (dash lne) for PVP/ethanol solutons under 25% RH (red) and 50% RH (blue). dameter at specfc length (d 3mm S,jet at 3 mm pont from the needle as well as d fnal S,jet at the end of straght jet regon). 5

6 At a crtcal length, perturbatons are not cancelled and the bendng regon starts. Ths can be characterzed by a bendng angle (θ) [10]. To measure the fber current (I), a current sensor s placed between the collector and NI USB-6008 data acquston (DAQ) card. Appled voltage, flow rate and RH are major controllable parameters n the electrospnnng process. To dentfy the allowable operatng bounds for a gven RH level and flow rate, voltage was vared from low to hgh n order to determne the upper and lower electrc feld bounds, between whch electrospnnng wll occur wth small varaton [10]. Below the lower voltage bound, there s a large varaton n Taylor cone volume and fallng droplets. Above the upper voltage bound, the Taylor cone s not observed, and the jet orgnates drectly wthn the needle and sometmes t moves around the edge of the needle tp. Ths procedure s repeated for dfferent flow rates to obtan the allowable operatng bounds dagram under a constant RH. The operatng bounds of solutons vary wth RH (fg.3), snce RH affects the jet current and the electrc force. We also are nterested n determnng how to relate operatng condtons and measurable process parameters to predct the fnal fber dameter. As descrbed n our early paper [14] for PEO/water solutons, the dameter of the upper straght jet dameter near the Taylor cone (d 3mm S,jet ) can be used as part of a correlaton to the fber dameter (d fber ). Snce lttle evaporaton occurs n the straght jet regon, the measured jet dameter s beleved to characterze the vscous-electrc force balance. The other factor used n the correlaton s proportonal to solvent evaporaton rate. A lnear correlaton s acheved (fg.4), f (1 RH) 2 s used. Ths correlaton suggests that there s a generalzable process physcs relatonshp, and also rases a queston of how does RH affect the fnal fber dameter? Snce alcohol s hydrophlc, the second order RH dependence suggests that water absorpton strongly affects the process physcs. 6

7 Fber dameter (nm) mm 0.3 mሶ evap (1 RH) 2 d S,jet Fg. 4 Correlaton of fber dameters to measurable parameters for PVP/alcohol solutons: (a) correlaton based on straght jet dameters; (b) correlaton based on straght jet dameters and solvent evaporaton rate; (c) correlaton based on straght jet dameter, solvent evaporaton rate as well as RH. Blue : PVP/methanol; Green : PVP/ethanol; Red: PVP/1-butanol; : 25% RH; : 35% RH; : 50% RH Fg. 5. Illuson of jet stretchng n straght jet regon and bendng regon. Fnal dry dameter based on the jet dameter at the end of straght jet regon reveals that stretchng s sgnfcant to determne fnal fber dameter n the bendng regon. (based on PVP/ethanol results) The role of the bendng regon relatve to straght jet regon n terms of stretchng and solvent evaporaton s llustrated n Fg. 5, based on PVP/ethanol expermental data (see appendx A of [12]). The d dry S,jet (fnal) s the jet dameter f the solvent were removed at the end of straght jet regon, and represents the fber dameter f there was no stretchng n 7

8 the bendng regon. d dry S,jet (fnal). Table.1 gves the values for the 3 solvent (4-10 μm), whch are much larger than the fnal fber dameters (1-3 μm). for PVP/methanol, PVP/ethanol and PVP/1-butanol solutons. Thus, the stretchng n the straght jet regon cannot explan the fnal fber dameter, and jet stretchng s sgnfcant n the bendng regon. The rato of the jet cross surface area change wth solvent removed n the straght jet regon (R straght = A dry S,jet (fnal) dry ) and n the bendng regon (R bendng = A S,jet (3mm) A S,jet A meas fber of jet stretchng occurs n the bendng regon for PVP/alcohol solutons. dry (fnal) ) show that ~51% In contrast, the degree of solvent evaporaton n the straght jet regon can be evaluated solvent by the rato of solvent evaporaton rate n the straght jet regon (mሶ S,evaporaton) over the jet solvent flow rate (Q (1 c)), as %Evap SJR = mሶ solvent S,evaporaton, where Q s the nfuse rate Q (1 c) ρ solvent of soluton; c s the ntal polymer concentraton; ρ s soluton densty; mሶ S,evaporaton solvent evaporaton rate n straght jet regon, whch s determned by the solvent evaporaton rate per area [13] and the jet surface area. The jet surface area s determned by the measured straght jet shape. Based on the solvent evaporaton rate, the calculated degree of solvent evaporaton n the straght jet regon for dfferent PVP/alcohol solutons, are all less than 1%. Thus, the bendng regon s sgnfcant n determnng the fber dameter, n terms of essentally all the evaporaton and half of the jet stretchng. s Table. 1 Degree of stretchng mpact on fber deformaton n bendng regon Materals PVP/methanol PVP/ethanol PVP/butanol dry d S,jet(3mm) (μm) d dry S,jet (fnal) (μm) d meas fber (μm) R straght = A dry S,jet(fnal) A dry S,jet (3mm) R bendng = meas A fber A dry S,jet (fnal) 4.9% 4.6% 11% 10% 14% 6.6% 8

9 3. Bendng regon model A model s developed, capturng the coupled mass and force balances, to get nsght nto the process physcs. Snce alcohol s a hydrophlc solvent, t absorbs water from the surroundng mosture ar, and both the alcohol solvent and the absorbed water must evaporate to acheve the fnal dry fbers. 3.1 Mass transfer: water absorpton and solvent evaporaton Due to the hydrophlc nature of alcohol, the mass transfer for a statonary alcohol lqud ncludes the water vapor flux (J water ar ), alcohol vapor flux (J solvent ar ) n the ar, and water the water dffuson (J soluton ), alcohol dffuson (J solvent soluton ) n the soluton. The water and alcohol fluxes n ar (J ar ) and the fluxes n soluton (J soluton ) are descrbed by an emprcal equaton (1) and Fck s frst law equaton (2) respectvely; J ar = h A (c surf (v) c ar (v)) Mw (1) J soluton = D A c (l) Mw (2) where h s the mass transfer coeffcent of water (solvent), whch s related to Sherwood number [6]; A s the surface area; Mw s molecular weght of water or solvent; D s the water dffusvty n alcohol; c surf (v) s the mole concentraton of water (solvent) vapor n the vapor phase; c ar (v) s the mole concentraton of water (solvent) vapor n the ambent ar; c (l) s the water (solvent) mole concentraton n lqud phase; c (l) s the water mole concentraton gradent n the lqud. Equaton (1) reveals that mass transfer n the ar s determned by the water or solvent vapor mole concentraton dfference between ambent ar (c ar ) and soluton surface n the vapor phase (c surf (v)). Assumng that the water vapor (c water ar ) and alcohol vapor (c solvent ar ) mole concentratons n ambent ar are constant (snce the amount of absorbed water and evaporated alcohol s relatvely small), the only unknown parameters for determnng water and solvent fluxes n ar (J ar ) are the water and solvent vapor mole concentratons on the surface. Equaton (2) ndcates that the dffuson n the soluton s drven by the water and solvent mole concentraton gradents ( c (l)) n the lqud, whch are the only unknown parameters to determne the water and solvent fluxes n soluton (J soluton ). 9

10 Thus, the only unknown parameters for the mass transfer are surface condtons, ncludng water and solvent mole concentratons and ther gradents. The relaton between c surface (v) and c surface (l) s gven by Raoult s law as: c surface (v) P = c surface (l) γ P saturated Mw soluton ρ ar (3) Mw ar ρ soluton where P s the ar pressure; P saturated s the saturated pressure of water (solvent) vapor n ar; γ s the correlaton for nteractons n lqud phase between dfferent molecules, called actvty coeffcent, whch s used to modfy the fractons or concentratons of components n a mxture. In a non-deal mxture, the mcroscopc nteractons between each par of chemcal speces are not same (e.g. the enthalpy change of soluton n mxng s not zero), and thus propertes of the mxtures cannot be expressed drectly n terms of smple concentratons or partal pressures. There are several methods to determne the actvty coeffcent of solvents systems. In ths paper, a UNIVERSAL Functonal Actvty Coeffcent (UNIFAC) model [15] s used to calculate the actvty coeffcent of each solvent where Mw soluton s the soluton molecular weght, Mw ar s the ar molecular weght, ρ soluton s the soluton densty and ρ ar s the ar densty. To determne the surface state, the dynamc of water (or solvent) mole concentraton on surface s calculated from the dfference n ar and soluton flows by: dc surface dt = J ar J soluton Mw A dh (4) where h s the soluton depth. To determne whch flux lmts the water absorpton for a statonary lqud, the relaton between resstances of the ar (R water water ar, equaton (5)) and lqud (R solutons, equaton (6)) mass transfer are evaluated, n order to calculate the mass transfer Bot number, (B m = water R soluton /R water ar ). The calculated B m for a statonary lqud wth 0.5 μ m s 4 (correspondng to the typcal fber dameter 1 μm), for 50 μm ts 328 (correspondng to the typcal upper jet dameter 100 μ m), and for 7 mm ts 45,900 (correspondng to the statonary alcohol lqud depth used n evaporaton experments dscussed later n ths water secton) depth. Although the B m decreases as lqud depth decreases, R soluton s much 10

11 larger than R water ar even when the lqud depth s only 0.5 μm, whch suggests that the water absorpton s lmted by the water dffuson n the soluton. water R solutons R water ar = 1 h w A Mw water (5) = h P water saturated γ water Mw soluton ρ ar (6) D A P ρ soluton Mw water Mw ar 2-D lumped model for an alcohol jet mass transfer n the bendng regon For a polymer/solvent jet, the mass transfer analyss must be extended to consder the jet shape. The water dffuson n the jet s analyzed n cylndrcal coordnate wth dfferent boundary condtons. The analytcal water concentraton n a jet wthout axal advecton (no soluton flow) s a functon of radus (r) and tme (t), expressed as: c water ~ ξ 1 e ξ 2 ξ 4 dξ, where ξ = r D t. Analyss of the analytcal soluton to the water concentraton profle as a functon of tme n a jet wth 5 μm radus under 50% RH, reveals that the absorbed water mole concentraton n the jet s not unform. Snce the water absorpton s lmted by water dffuson n the jet (B m 1), t s requred to determne the water mole concentraton profle n the jet to capture the water dffuson. A lumped model to descrbe the mass transfer n jet usng a set of 3 radal lumps ( R 1 : R 2 : R 3 = 5:4:1 from jet surface to jet central) are used. The modeled water concentraton profle s compared to the analytcal water concentraton profle, and ndcates that the 3 radal lumps captures the water dffuson n the jet [12]. To capture the axal advectve flux of water and alcohol due to the jet soluton flow and the radal dffusve flux of water and alcohol cross the jet, a 2-D lumped model (fg. 6) s used to determne the water and alcohol mole concentraton profles and the mass transfer rates for a n-th set of jet radal lumps wth 1mm lump length. The determnaton for the lump length s gven n appendx D of [12]. 11

12 n-th set of radal lumps water J 1 m,n 2 solvent J 1 m,n 2 water J ar,n water J surface,n water J 1 m+ 2,n water J 1 m 2,n c surf,n (v) c surf,n (l) m+1, n c m+1,n m, n c m,n m-1, n c m 1,n C L ethanol J ar,n r solvent J surface,n solvent solvent J 1 J m+ 2,n 1 m 2,n s For a set of 3 radal lumps secton, there are eght unknown parameters whch are water and solvent mole concentratons on surface ( c surf,n concentratons n each of the 3 radal lumps (c m+1,n, c m,n ) and water and solvent mole, and c m 1,n ). The unknown lump states are determned by teratvely solvng the mass balance equatons n each lump and on surface untl a convergence s acheved (see chapter 7 n [12]). Once the lump states are determned, the mass fluxes n terms of evaporaton and absorpton of water and alcohol surface, the radal dffuson and axal advecton are determned based on the fluxes functons. The evaporaton and absorpton of alcohol and water on surface are two sgnfcant fluxes we are nterested n, snce they determne how much water s absorbed, the total axal jet flow rate, and how fast the jet soldfes. Once the water and solvent mole concentratons on the n-th set of 3 radal lumps surface (c surf,n (v)) are calculated, the water absorpton rate and alcohol evaporaton rate on n-th set of 3 radal lumps surface (equaton (1)) can be determned from: water J 1 m,n+ 2 solvent J 1 m,n+ 2 Fg. 6 Sketch llustratng two-dmensonal numercal analyss of water absorpton and solvent evaporaton for n-th set of radal lumps 12

13 mሶ n water water = J ar,n = h w A n (c water surf,n (v) c water ar ) Mw water = 1.95v 6 3 a v ar R 3 n (c water surf,n (v) RH P water 1 saturated ) D 2 Mw water ar water ds (7) R ar T mሶ n solvent solvent = J ar,n = h s A n (c solvent surf,n (v) c solvent ar ) Mw solvent = 1.95v 6 3 a v ar R 3 n c solvent 2 surf,n (v) D solvent ar Mw solvent ds (8) where v a s the knematc vscosty of ar; ds s the dfferental jet length; R ar s deal gas constant; T s the temperature; v ar s the ar cross velocty, 10m/s, whch s determned n [6]. Equaton (7) reveals that the water wll start to evaporate from the jet surface, when the water vapor mole concentraton on the n-th set of 3 radal lumps surface s larger than the ambent water vapor mole concentraton. By assumng that the solvent vapor mole concentraton n ar s 0, and both water and solvent vapor concentratons do not change n the ar concentratons (snce the amount of absorbed water and evaporated alcohol s relatvely small), the net evaporaton rate for the n-th set of 3 radal lumps s: mሶ net,n = dq ρ soluton = mሶ n solvent + mሶ n water (9) whch s a functon of jet flow rate (Q) and jet radus (R). 3.2 Force balance In the bendng regon, the dfferental equaton (10) for force balance s bult on the work of Hohman [1, 2] and Feng [3]. To numercally solve the force balance dfferental equaton for the bendng regon, the bendng regon jet s cascaded nto dscrete sets of 3 radal lumps wth a short length ds. The force balance performed on each node requres the net force actng on the set of 3 radal lumps s equal to the change of momentum that fluxes n and out the set of radal lumps. For the n-th set of 3 radal lump n the bendng regon, although the water and alcohol mole concentraton profles n the 3 radal lumps are not unform, the average lump parameters (e.g. polymer concentraton, vscosty and jet flow velocty) are assumed to be unform n the force balance equaton. 13

14 To determne the change n jet shape, a force balance s developed from a steady state momentum equaton for each lump n the bendng regon. Our prevous analyss [6] has dentfed the domnant force terms n the bendng regon, ncludng the extensonal vscous force (F vs), charge-to-charge force (F charge) and electrc feld force (F e-feld) (fg. 7). The steady state momentum equaton n the bendng regon s expressed: ρv s d πr 2 (Q) + ρv d ds s (v ds s) = 3 d R 2 ds (R2 η d v ds s) + [ σ s d σ ε ds s 2σ s d (Rσ ε R ds s)] + 2σ se R ( 1. DM ) (2. Fvs) (3. Fcharge) (4. Fe-feld) Dt snα (10) where DM Dt s materal dervatve of momentum; jet flow velocty (v s) s determned by jet Q flow rate and jet radus, v s = πr 2; Surface charge densty (σ s) s a functon of measured jet current, and jet radus, σ s = IR 2Q [7]. In the bendng regon, equaton 10 ndcates that the momentum change equals to the sum of the extensonal vscous retardng force (term 2) and the electrc stretchng force n terms of electrc feld force (term 4) and charge-to-charge force (term 3). The steady state momentum equaton s consstent wth the work of Hohman [1] and Feng [3] except for addton of 3 factors: a) the helx angle (α) n terms of the electrc feld force (4. F e feld ), b) the charge-to-charge force (3. F charge ) term s expanded to nclude the axal charge-to- axal charge force (F charge = 2σ s ε R d ds (Rσ s)), whch s a sgnfcant stretchng force n the bendng regon. (It s noted that the addtonal axal term n the charge-to-charge force acts opposte to the term gven n Feng s equaton, and thus, acts to stretch the jet) c) the effect of evaporaton on the change of momentum (Mሶ = ρv s d (Q) + ρv d πr 2 ds s (v ds s)). In the straght jet regon, snce evaporaton s neglgble ( d Q = 0), the momentum change on the left ds hand sde can be reduced to Mሶ = ρv s d ds (v s), whch s consstent wth Feng s equaton [3]. In the bendng regon, the momentum change due to the evaporaton s also sgnfcant. 14

15 Mሶ Fvs R b Bendng axs Fcharge+Fe-feld α Horzontal axs Fg. 7. Force balance for a set of radal lumps n the bendng regon The force balance can be rewrtten as a functon of jet radus (R) and jet flow rate (Q) n order to explctly ndcate how the jet radus vares n terms of change of forces along the jet, by couplng the mass and force balances: [ 2ρQ π 2 R 4 dq ds 2ρQ2 dr π 2 R 5 ds ] = [ 6η dr πr 3 ds dq + 3η d 2 Q + 6η Q ds πr 2 ds 2 πr 4 (dr ds )2 6η Q d 2 R πr 3 ds 2] + [ 3I2 R dr + I2 R 2 dq ] 4εQ 2 ds 4εQ 3 ds (1. Mሶ ) (2. F vs) (3. F charge) +[ E I sn α ] (11) Q (4. F e-feld) where I s measured jet current; ε s permttvty of ar; E s the ambent electrc feld; α s the helx angle [6]; Q s volume flow rate along the jet; and η s the extensonal vscosty. The force and momentum terms affected by solvent evaporaton rate are hghlghted n equaton (11). The extensonal vscous force (term 2. Fvs n equaton (11)) s affected by change n the extensonal vscosty (η ) and the net evaporaton rate ( dq ). [12]. The net evaporaton ds rate s determned by the mass transfer/ evaporaton rate terms, equaton (9). The extensonal vscosty (η ) s related to the non-dmensonal parameter Trouton rato (Tr) to the shear vscosty (η 0 ), by η = η 0 Tr [6], where η 0 s a measured shear vscosty and changes wth polymer concentraton. Snce we have no way to measure Tr, we use a ftted Trouton rato, Tr, by comparng modelng results and expermental observatons. 15

16 The ftted Trouton rato, Tr, s determned by comparng model results to the observatons of the jet dameter change rate ( dd ) at the end of straght jet regon and fnal fber dameter. ds A value Tr ~10 s found to work well for PVP/methanol and PVP/ethanol solutons, and a value of Tr ~100, s found to work well for PVP/1-butanol solutons. These ftted Trouton rato values acheve good modelng results over a broad range of operatng condtons, n terms of flow rate and RH (whch sgnfcantly affects the evaporaton rate and thus Q s ), whch results n dfferent extensonal vscous force condtons [12]. In the bendng regon, the shear vscosty (η 0 ) ncreases as the polymer concentraton ncreases due to solvent evaporaton, but also changes due to the relatvely smaller water mole concentraton change due to water absorpton. The measured soluton shear vscosty s shown as a functon of polymer concentraton [12] and yelds: η 0 ~c 2.65 for PVP/methanol, η 0 ~c 2.67 for PVP/ethanol, η 0 ~c 3.58 for PVP/1-butanol. 3.3 Expermental verfcaton To verfy the model, model and expermental results were compared for a varety of PVP/alcohol solutons conducted over a broad range of operatng condtons and soluton propertes. In partcular, flow rate vared by a factor of 2.3, voltage vared from the lower to upper bounds, RH vared by a factor of 2, soluton shear vscosty vared by a factor of 6.3, conductvty vared by a factor of 6.6, solvent evaporaton rate vared by a factor of 11.3, and water absorpton rate vared by a factor of Ths resulted n a large range of the fnal fber dameter, whch vared by a factor of 7. Fg. 8 shows the comparson of molded results to expermental fber dameters for the dfferent RH and solvents, whch shows that RH has a relatvely larger mpact than flow rate and voltage. The average absolute fber dameter error s less than 8% for ths a broad range of operatng condtons and soluton propertes. Ths suggests that the developed model captures the domnant process physcs n the bendng regon for non-aqueous hydrophlc PVP/alcohol solutons. 16

17 Fbet dameter (nm) In order to understand the mpact that the range of soluton propertes have on the experments, the terms of the force balance (equaton 11) and mass balance (equaton 9) are used to evaluate the mpacts expressed n terms of forces and jet length. Snce ~ 90% jet dameter thnnng occurs n the transton regon and the lower bendng regon (fg. 10), the forces n terms of extensonal vscous force, charge-to-charge force and net stretchng force are characterzed at the md-pont of the transton regon and lower bendng regon for each PVP/alcohol soluton case. To understand the relatve mpact, the forces are normalzed by the electrc force of PVP/methanol solutons, whch has the largest stretchng force n those PVP/alcohol soluton cases. The jet length s normalzed by the calculated jet length of PVP/1-butanol soluton, whch has smallest net evaporaton rate and thus, has the longest jet length % RH for PVP/1-butanol 50% RH for PVP/1-butanol mm 0.3 mሶ evap (1 RH) 2 d S,jet Fg. 8 Comparson expermental measurements (blue) and modelng predctons (red) of fber dameters for PVP/methanol ( ), PVP/ethanol ( ), and PVP/1-butanol ( ) solutons under dfferent operatng condtons (upper and lower voltage bounds for RH: 25% - 50%, Q: ml/mn). The absolute average fber dameter error s less than 8% Fg. 9 shows the normalzed jet length, extensonal vscous force, electrc stretchng force, and the net force for the 3 PVP/alcohol soluton cases and t s nterestng to compare these results to the range of property values (fg. 1). Snce the bendng regon jet length s determned by net evaporaton rate, PVP/1-butanol has a longer jet length (4.8 tmes) than PVP/ethanol due to ts smaller net evaporaton rate (7 tmes). Comparson of the soluton propertes to the forces ndcates that soluton propertes are not suffcent to determne 17 25% RH for PVP/1-butanol

18 forces. Although PVP/1-butanol has a much larger shear vscosty (7.9 tmes) than PVP/methanol, the extensonal vscous force of PVP/1-butanol s 10.6 tmes larger than that of PVP/methanol, snce the extensonal vscous force s also affected by Trouton rato and net evaporaton rate (equaton 11). In addton, although a larger conductvty leads to a larger current, the conductvty s not suffcent to determne electrc force, whch s determned n part by charge densty and electrc feld (equaton 11). PVP/1-butanol has a smaller (15 tmes) conductvty than PVP/methanol, whch results n a smaller (4 tmes) electrc force. For PVP/alcohol solutons, snce PVP/1-butanol has the smallest electrc force and the largest electrc force, t has the smallest net stretchng force. It s also notced that the extensonal vscous forces of PVP/methanol and PVP/ethanol are much smaller (~ 10 tmes) than the electrc forces for them due to ther relatvely large conductvty and small vscosty. Thus, for PVP/methanol and PVP/ethanol, the extensonal vscous force s not mportant. However, the extensonal vscous force and electrc force for PVP/1-butanol are at the same order of magntude. Extensonal vscous force Net force PVP/ethanol PVP/1-butanol Jet length PVP/methanol Electrc force Fg. 9 Normalzed extensonal vscous force, electrc force and net force, as well as jet length for PVP/methanol, PVP/ethanol and PVP/1-butanol solutons (0.05 ml/mn and 35% RH) 18

19 Jet dameter (um) Net stretchng force (N) 4. Model analyss Ths secton analyzes the role of water absorpton, RH, alcohol evaporaton and jet stretchng n determnng the resultng fber dameter usng the modelng results of PVP/ethanol solutons. PVP/ethanol solutons have forces n-between those of PVP/methanol and PVP/1-butanol, but has the largest net evaporaton rate and therefore the shortest jet length. The detaled modelng results for PVP/methanol and PVP/butanol are gven n chapter 8 n [12]). 4.1 Contrbutons of mass transfer and stretchng The modeled jet dameter and net force for PVP/ethanol n the bendng regon s shown n fg. 10, llustratng the degree of stretchng and evaporaton. The jet dameter wth solvent removed represents the contrbuton of jet stretchng, and shows that ~28% of dameter change n the bendng regon s due to stretchng. Although the contrbuton of stretchng s smaller than that of net evaporaton, the bendng regon jet stretchng s mportant snce the jet dameter only decreases to nano-scale at the late stage of bendng regon E-08 3E E-08 2E E-08 Stretchng 1E-08 contrbuton 5E-09 1 μm 0 s md Bendng jet length (m) Upper bendng regon: 10% Transton regon: 18% Lower bendng regon: 72% Evaporaton contrbuton Fg. 10 Model predcton of jet dameter (red), jet dameter wth solvent removed (green), and net stretchng force (blue) n the bendng regon for PVP/ethanol solutons (35% RH, 0.05 ml/mn and 12 kv). 19

20 Comparng the shape of the net stretchng force and jet dameter curves (fg.10) reveals that although the net force s very small through most the length of the jet n the bendng regon, t has a sgnfcant mpact n terms of jet stretchng. The bendng regon can be separated nto 3 regons: upper bendng regon wth large net force decrease, transton regon and lower bendng regon wth a small net force. Although the length n the upper bendng regon s short (~0.1 m whch s 2.3% of the bendng jet length), the jet dameter decreases 10% due to the large net stretchng force. However, 72% of the jet dameter decreases n the lower bendng regon due to ts long jet length (~ 3.5 m whch s 81% of the bendng jet length), even though the net force s ~ 3 orders of magntude smaller. Comparng the jet shape and net stretchng force (fg.10) wth the net evaporaton rate curve (fg. 11(a)) reveals that the jet thnnng rate follows the evaporaton rate. Fg. 10 shows that the jet dameter has a sharp but lmted decrease n the upper bendng regon due to the large net stretchng force. Ths observaton s smlar to the measured jet shape n the straght jet regon (see appendx A n [12]). However, n the straght jet regon, evaporaton s neglgble, the jet dameter decreases dramatcally ~ 160 tmes due to the large net stretchng force [11] over a short porton of the straght jet length, and then slowly decrease. In contrast, the jet dameter decreases at lmted levels for much of the bendng regon. Fg. 11(a) shows that the net evaporaton rate decreases slowly n the bendng 1/3 regon snce the evaporaton rate only s weakly determned by jet shape, mሶ net,n~r n (equaton (9)). Ths results n a lmted decrease n jet dameter n the transton and lower bendng regons. In summary, the large change of net stretchng force n the upper bendng regon has a relatvely small effect on jet thnnng rate relatve to the effect of net evaporaton rate. 4.2 Effect of RH on mass and force balances To determne how RH affects the electrospnnng process and fnal fber dameter for PVP/alcohol solutons, the jet behavor for 3 dfferent RH cases for PVP/ethanol solutons are analyzed. The expermental condtons and modelng results for those 3 cases are gven n table. 3 whch show the correspondng jet behavor for the bendng regon. As RH ncreases from 25% to 50%, the jet length ncreases ~20% and the fber dameter decreases ~25%. 20

21 Table. 3 Expermental measurements and modelng results for PVP/ethanol solutons under 3 dfferent RH levels RH Q (ml/mn) V (kv) I (na) 3mm d S,jet (um) d S,jet (fnal) (um) Calculated L jet (m) Measured d fber (nm) The plot of net evaporaton rate as a functon of jet length as a functon of RH (fg. 11(a)) shows that a hgher RH leads to a decrease n net evaporaton rate. A decrease of net evaporaton rate results n a longer jet length and stretchng tme. Why does net evaporaton rate decrease wth hgher RH? The mole fracton of ethanol on the jet surface decreases for greater RH snce the mole fracton of absorbed water ncreases (fg. 11(b)). Ths results n the net evaporaton rate decreasng for greater RH snce the decrease n alcohol evaporaton rate are greater than the amount of water absorpton as shown n fg. 11(c). Calculated d fber (nm) 50% % %

22 Surface mole fracton n lqud phase Net evaporaton rate (g Τ (m 2 s) ) (a) Ethanol Water (b) Ethanol Water (c) Fg. 11 (a) modelng net evaporaton rates for PVP/ethanol solutons under dfferent RH condtons; (b) modelng water and ethanol mole fractons on surface for dfferent RH condtons; (c) modelng water absorpton rate and ethanol evaporaton rate for dfferent RH condtons. (Red: 25% RH; green: 35% RH; blue: 50% RH, postve: evaporaton, negatve: absorpton) 22

23 When does the absorbed water start to evaporate from the jet? When the water mole fracton on surface s larger than that n the ar, the water starts to evaporate from the jet whch occurs at ~ 3m. Fg. 11 (c) also shows that the water absorpton rate s ~ 10 tmes smaller than the solvent evaporaton rate on the jet surface, whch occurs snce water absorpton s lmted by dffuson (B m 1) and solvent evaporaton rate s determned by the solvent mole concentraton on surface. Thus, the major reason for the decrease of net evaporaton rate wth ncreasng RH s the decrease of ethanol evaporaton rate due to larger amount of water occupyng the surface states. Although the amount of absorbed water s small relatve to the solvent, RH sgnfcantly affects the net evaporaton rate snce RH sgnfcantly changes the surface state dstrbuton whch ultmately determnes the solvent evaporaton rate (equaton (8)). The net stretchng force s also found to ncrease as RH ncreases, as shown n fg. 12. At 3m (whch s ~70% of bendng jet length), the calculated net stretchng force ncreases a modest ~7% as RH ncreases from 25% to 50%. The ncrease of net stretchng force wth ncreasng RH s a result of the decrease n both electrc stretchng and extensonal vscous retardng forces. Snce the extensonal vscous retardng force decreases by ~40%, whch s larger than the decrease (~4.7%) n electrc stretchng force (fg. 13), the net stretchng force ncreases as RH ncreases (fg. 12). 50% F net 25% = 1.07 F net Fg. 12 modeled net stretchng forces for PVP/ethanol solutons under 25% RH (red), 35% RH (green) and 50% RH (blue). As RH ncreases from 25% to 50%, the calculated net stretchng force ncreases ~ 7% at 3 m n the bendng regon (upper bounds of 0.05 ml/mn). 23

24 Overall, both the total jet length and net stretchng force determne the fnal fber dameter. Understandng the magntude of the effect that RH has on the net stretchng force and jet length s mportant for understandng the overall mpact of RH. For PVP/ethanol solutons, as RH ncreases from 25% to 50%, although the extensonal vscous force decreases ~40%, the net stretchng force only ncreases a modest ~7% snce the electrc force s much larger (~10 tmes) than the extensonal vscous force due to the relatvely small vscosty and large conductvty (fg. 1). However, ths ncrease n the net stretchng force cannot explan the ~25% decrease n fber dameter (fg. 10) tself. The major factor s the ~20% ncrease n jet length (whch ncreases the stretchng tme). Whle PVP/methanol has smlar results (see chapter 8 n [12]), PVP/1-butanol behaves sgnfcantly dfferent snce the electrc force has a smlar magntude on the extensonal vscous force due to the relatvely large vscosty and relatvely small conductvty (fg. 1). For PVP/1-butanol, as RH ncreases from 25% to 50%, the net stretchng force ncreases 45% due to the 31% decrease n extensonal vscous force and 8% decrease n the electrc force. The overall 60% decrease n fber dameter results from the 45% ncrease n net stretchng force and 20% ncrease n jet length. F e 25% F e 50% = ~1.05 Electrc force 25% F vs 50% = ~1.4 F vs Extensonal Vscous force Fg. 13 Plot of vscous retardng and electrc stretchng forces as a functon of RH (red: 25% RH, green: 35% RH, blue: 50% RH). Net stretchng force ncreases snce reducton of extensonal vscous force wth ncreasng RH s greater than that of electrc force. 24

25 The model analyss provdes nsght as to what drves the change n force as RH changes. Both the electrc feld force and charge-to-charge force vary wth the surface charge densty (σ s, see equaton (10)). As RH ncreases, the net evaporaton rate decreases, whch results n a decrease n surface charge densty (fg. 14), and thus the electrc force, snce the surface charge densty s a functon of jet flow rate (equaton (11)). The extensonal vscous force (equaton (10)) s determned by shear stress (τ = 6η R dv s dr + ds ds 3η d2 v s ds2 ) [6]. Thus, as RH ncreases, the net evaporaton rate decreases, whch results n a smaller dv s and d2 v s ds ds2, and thus shear stress (fg. 15) and extensonal vscous force decrease. Fg. 14 Plot of modelng surface charge densty for PVP/ethanol solutons along the jet for 25% RH (red), 35% RH (green) and 50% RH (blue). Fg. 15 Plot of modelng shear stress for PVP/ethanol solutons along the jet for 25% RH (red), 35% RH (green) and 50% RH (blue). 25

26 5. Summary In ths paper, a combned expermental and modelng analyss of the bendng regon for 3 dfferent non-aqueous hydrophlc PVP/alcohol solutons s presented. Expermental observatons reveal that both jet stretchng and evaporaton n the bendng regon s sgnfcant n determnng the fnal fber dameter. To obtan nsght nto the process physcs, a model s developed that captures the coupled mass and force balances, and predcts the fnal fber dameter to wthn 8% of expermental results over a broad range of operatng condtons and soluton propertes. The mass balance ncludes the evaporaton/ absorpton of alcohol and water on surface, the radal dffuson of alcohol and water n jet, and the axal advecton of alcohol and water n jet. Water dffuson n the jet s found to lmt the absorpton rate, and the speces absorpton and dffuson model s expermentally verfed. The force balance s based on the domnant factors ncludng extensonal vscous force and the electrc stretchng forces (electrc feld and charge-to-charge forces). Model analyss reveals that the net evaporaton rate (sum of solvent evaporaton rate and water absorpton rate on surface) has two sgnfcant mpacts. Frst, the net evaporaton rate affects the jet length and the stretchng tme, wheren the longer the jet, the smaller the fber dameter. Secondly, the net evaporaton rate also affects the net stretchng force, snce the mass flow affects the charge densty and vscous terms. The effect of net evaporaton rate on extensonal vscous retardng force s greater than the mpact on the electrc stretchng force. For PVP/methanol and PVP/ethanol solutons, snce the extensonal vscous force s much smaller than the electrc force, the effect of RH on net stretchng force s relatvely small. However, for PVP/1-butanol solutons, the effect of RH on net stretchng force s sgnfcant. The larger RH results n a larger net stretchng force and a smaller fber dameter. Analyss of the mpact of RH reveals that although the amount of absorbed water s not large, t has a profound mpact on the solvent evaporaton rate and thus fnal fber dameter. As the ambent RH ncreases, the water mole concentraton on the surface ncreases, resultng n a decrease n alcohol mole concentraton, whch leads to a lower solvent evaporaton rate and a hgher water absorpton rate. The decrease of net evaporaton rate results n a longer jet and larger net stretchng force. Thus, the fnal fber dameters of nonaqueous hydrophlc PVP/alcohol solutons decrease as RH ncreases. 26

27 Usng the model, the mpact of dfferent soluton propertes s determned n terms of the relaton between forces and evaporaton rate to the resultng fber dameter. Acknowledgements We apprecate the fundng support from the Army (W911QY ), and the contrbutons of Jnjang Lu n conductng experments. We also acknowledge the contrbuton of that Keth Forward made dscusson of the mass transfer model development. Reference [1] Hohman, M.M., Shn, M., Rutledge, G. and Brenner, M.P., Electrospnnng and electrcally forced jets. I. Stablty theory. Physcs of Fluds (1994-present), 13(8), pp [2] Hohman, M.M., Shn, M., Rutledge, G. and Brenner, M.P., Electrospnnng and electrcally forced jets. II. Applcatons. Physcs of Fluds (1994-present), 13(8), pp [3] Feng, J.J., The stretchng of an electrfed non-newtonan jet: A model for electrospnnng. Physcs of Fluds (1994-present), 14(11), pp.39e [4] Rutledge, G.C. and Frdrkh, S.V., Formaton of fbers by electrospnnng. Advanced Drug Delvery Revews, 59(14), pp [5] De Vreze, S., Van Camp, T., Nelvg, A., Hagström, B., Westbroek, P. and De Clerck, K., The effect of temperature and humdty on electrospnnng. Journal of materals scence, 44(5), pp [6] Ca, Y. and Gevelber, M., Analyss of bendng regon physcs n determnng electrospun fber dameter: effect of relatve humdty on evaporaton and force balance. Journal of Materals Scence, pp [7] Wang C.W, Hsu C.H, and Ln J.H. Scalng laws n electrospnnng of polystyrene solutons. Macromolecules, 39:7662{7672, [8] Yarn, A.L., Koombhongse, S. and Reneker, D.H., Bendng nstablty n electrospnnng of nanofbers. Journal of Appled Physcs, 89(5), pp [9] Forward, K.M., Flores, A. and Rutledge, G.C., Producton of core/shell fbers by electrospnnng from a free surface. Chemcal Engneerng Scence,104, pp [10] Yan X. "Electrospnnng of nanofbers: analyss of dameter dstrbuton and process dynamcs for control. " Thess (Ph. D.) Boston Unversty, [11] Ca Y. "Electrospnnng process analyss: the relaton of process parameters to fber dameter and process dynamcs for closed-loop control desgn. " Thess (Master) Boston Unversty, [12] Ca, Y., Modelng and expermental analyss of electrospnnng bendng regon physcs n determnng fber dameter for hydrophlc polymer solvent systems (Doctoral dssertaton, Boston Unversty). 27

28 [13] Smallwood, I., Handbook of organc solvent propertes. Butterworth- Henemann. [14] Ca, Y. and Gevelber, M., The effect of relatve humdty and evaporaton rate on electrospnnng: fber dameter and measurement for control mplcatons. Journal of Materals Scence, 48(22), pp [15] Fredenslund, A., Vapor-lqud equlbra usng UNIFAC: a group-contrbuton method. Elsever. 28