User s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities

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1 File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear Least-Squares Fits with a Common Interept: Determination of the Intrinsi Visosity of Maromoleules in Solution M.C. López Martínez, F.G. Díaz Baños, A. Ortega and J. Garía de la Torre Journal of Chemial Eduation 80, (2003) A ontribution from: Departamento de Químia Físia, Universidad de Muria, Muria, Spain The latest versions of the VISFIT program and related doumentation an be found at: 1. Introdution The intrinsi visosity, [], is one of the most useful properties of maromoleules in solution. It an be employed to determine the moleular weight of the maromoleule, provided that the parameters K and a in the Mark-Houwink- Sakurada equation, []=KM a are known. For flexible hain maromoleules, it an provide information on the dimensions of the maromoleular oil and the flexibility of the hain skeleton. In the field of biologial maromoleules, it is well known that it depends very sensitively on the overall struture, adopting values for elongated strutures that are muh larger than those proper of globular shapes. The inrease in the solution visosity, with respet to that of the pure solvent 0, due to the presene of the maromoleular solute, with onentration, an be measured by the relative visosity. 1

2 = r (1) 0 or by the speifi visosity 0 sp = = r 1 (2) 0 The intrinsi visosity [] is defined as the ratio of speifi visosity to onentration in the limit of infinite dilution: sp 0 r 1 [ ] = lim = lim = lim (3) The dependene of solution visosity on solute onentration at the infinite dilution limit an be expressed in several ways. At infinite dilution, sp / equals [], and for moderately diluted solution the onentration dependene an be expressed in a linear form, the Huggings equation [ ] + k [ ] = (4) sp 2 H so that [] ould be obtained by extrapolation to zero onentration of data for sp / at varying onentration. Typially, this would be done by a least-squares fitting, in whih the interept is [] and the slope is k H [] 2 It an be shown that [] is also the zero-onentration limit of ln r /. Then, Kraemer proposed to express the onentration dependene of this quantity as: ln ( r ) 2 [ ] k [ ] = (5) K Thus, the Kraemer extrapolation has the same interept [] as the Huggins equation. It an be shown that k H +k K =1/2 (6) This relationship omes from purely mathematial arguments and must hold for any polymer / solvent system. Rather than doing a separate linear least-squares extrapolation for eah of the two relationships, eqs. 4 and 5, we propose to fit simultaneously both equations. We have devised a least-squares proedure whih takes into aount that the two series of data have the same abisas and a ommon interept. The mathematial details (desribed in our above ited paper) were implemented in a omputer program, VISFIT, whih is intended for handling the experimental visosity-onentration data in the determination of the intrinsi visosity. 2

3 2. Experimental proedure and VISFIT operating modes. The determination of onentration-dependent solution visosities is most easily arried out using a dilution Ubbelohde visosimeter, whih differs from the lassial Ostwald visosimeter in having a third arm employed to add amounts of pure solvent, so that hange in onentration an be made in situ. The visosity is determined from the time, t, required for a given volume of solution to flow falling through a apillary. First of all, a measurement of the falling time of the pure solvent, t 0, is done. Then, putting in the visosimeter a volume V 1 of a solution of onentration 1, the falling time t 1 is measured. Afterwards, a series of new measurements is done adding a volume δ i of pure solvent and obtaining the new falling time t i, whih orresponds to a onentration related to the previous one by i = i-1 V i-1 /V i = i-1 V i-1 /(V i-1 + δ i ). Thus the primary set of data is a series of pairs of values (δ i, t i ). The flow rate aross the apillary is governed by the Hagen-Poiseuille law, whih states that the flow time (for a given volume) is proportional to the density of the fluid, ρ and inversely proportional to its visosity. Aordingly, the visosity an be writen as = A ρ t, where A is an instrumental onstant depending on onstrution details of the visosimeter. This is approximately valid, but a better desription is ahieved taking into aount end effets arising from the fat that the veloity profile at the extremities of the apillary is not Pouiseuillean. Then, the visosity is given by ( At B / t) = ρ (7) where B is another instrumental onstant. The visosimeter an be alibrated, determining the values of A and B from a series of measurements of flow time for pure liquids of known density and visosity, at the same temperature as the solution measurements. 2.a. VISFIT (simple) mode 1 Our VISFIT omputer program for the determination of the intrinsi visosity has two operational modes. In the approximate (but still aeptably preise) mode 1, end orretions (the B onstant) is negleted, and furthermore the solution density is supposed to be the same as that of the pure solvent (the latter approximation is quite aurate for the usual, quite dilute solutions employed in the experiment). Therefore, we are in the ase when = A ρ t. As all we need are values of r (or sp = r -1), they an be simply alulated as the ratio of flow times, r = t / t 0, where t 0 is the flow time of the pure solvent. Thus in this mode neither densities nor the A and B onstants are needed. 3

4 2.b. VISFIT (advaned) mode 2 In the more rigorous mode 2 of VISFIT, both the density variation and the apillary end effets are taken into aount. The latter are represented by the visosimeter onstants A and B, obtained from a separate, previous experiment in whih the visosimeter was alibrated as mentioned above. For the onentration-dependent density, a linear relationship is assumed to be valid in dilute solution. If ρ 1 is the solution density (determined apart, for instane by pinometry) at onentration 1 (whih ould be that of the initial, most onentrated solution), and ρ 0 is the density of pure solvent, then at any onentration the density of the solution an be estimated as ρ = ρ 0 + (ρ 1 -ρ 0 )/ 1. In this version of VISFIT, the program will need data of A and B, ρ 0, ρ 1 and 1. The instrumental orretion may be more important than that from the hange in density. If one would like to inlude the former, ignoring the latter (thus avoiding the measurements of density), this ould be done giving ρ 0 for ρ 1 (indeed the value given will be irrelevant; it will anel out in the evaluation of r ). Similarly, one ould neglet the alibration orretion, still inluding the densities; this is done setting B = 0 and giving any arbitrary value to A. 3. Running VISFIT The input data for VISFIT an be either (a) given from the keyboard, following the program s prompts, or (b) supplied in an input file. In the latter ase, the only line to be typed is that with the name of the input file. The data set ontains: Name of the output results file (*) Name of the output graphis file (*) Title of the ase (e.g., polymer/solvent/temperature) (*) Flow time for the pure solvent Conentration of the initial solution (before dilution) Volume of the initial solution Flow time of the initial solution Operating mode (1-basi, 2-advaned). If mode is 2, advaned, three more line will follow: Values (omma separated) of the onstants A and B (omma separated) of the apillary visometer (in gs units) Solvent density Density of the initial solution Finally there will be a series of lines, one for eah flow time measurements, giving Values (omma separated) of added solvent volume and resulting flow time Values (omma separated) of added solvent volume and resulting flow time... et -1., -1. Indiates end of data set (*) In the input data file, these three names will have 30 haraters, so add the needed spaes after the last letter of the name 4

5 An example of the input file is presented below. VISFIT will generate a main results file, also given below. Finally, the program produes a file intended to the exported to a graphis tool (we employ SigmaPlot). This file ontains, first, three olumns with the experimental values of, sp / and ln rel /, whih define the two sets of data points : (a) (, sp /), (b) (ln rel /) Similarly, the other three olumns ontain two lines with, sp / and ln rel / values. The first line is for = 0, and the two aompanying numbers are the same as the intrinsi visosity. The other line orrespond to a onentration slightly larger than the initial one, and orrespond to points (, sp /), (ln rel /), that when joined with the ommon interept give the two fitted straight lines. 4. Example The following is an example of the input file of VISFIT in the advane mode, taken from a laboratory experiment. Polystyrene (Aldrih, 43,010-2), was dissolved in toluene (Fluka, 89675). Chemials were used without further purifiation. Solution and solvent were filtered through a filter funnel. The visosity measurements were determined using an Ubbelhode visosimeter onstruted in our glass-blowing shop, immersed in a home-made thermostated bath of transparent polymethylmetarylate walls, with a temperature ontrol preise within ± 0.05ºC. The measurement of the falling time was done with an inexpensive digital stopwath. The input file for VISFIT is listed in Figure 1. outputres-2.txt!name of results file (30 har.) outputgra-2.txt!name of file for graphis (30 har.) Polystyrene/Toluene/25.0 C!Title for experiment (30 har.) 41.2!Flow time of pure solvent (s) !Conentration of initial solution (g/m^3) 15.!Volume of initial solution (m^3) 76.30!Flow time of initial solution (s) 2!MODE: 1-Elementary 2-Advaned , 0.029!Visosimeter onstants, A (m^2 s^-2), B (m^2) !Solvent density (g/m^3) !Initial solution density (g/m^3) 1.0, 73.77!Added volume of solvent (m^3), and flow time (s) 3.1, 67.60!Added volume of solvent (m^3), and flow time (s) 2.9, 63.46!Added volume of solvent (m^3), and flow time (s) 8.0, 56.73!Added volume of solvent (m^3), and flow time (s) 10., 52.32!Added volume of solvent (m^3), and flow time (s) -1., -1.!End of data Figure 1. Input data file for VISFIT in the advaned mode. If we had operated in the simplest mode 1, whih do not require a previous alibration and the onentration dependene of density, then in the input file the lines with A, B, ρ 0, ρ 1 and 1 would be missing. VISFIT produes a main output file, displayed in Figure 2, with the results of the fit, inluding the intermediate alulations for eah onentration, and the results of the 5

6 double least-squares fit. The values of the intrinsi visosity and the Huggins and Kraemer onstants are finally reported. Title: Polystyrene/Toluene/25.0 C Name of output graphis file:outputgra-2.txt Flow time for solvent (s): Conentration of initial solution (g/m3): Volume of initial solution (m3): Flow time of initial solution (s): Visosimeter onstants A (m^2 s^-2) and B (m^2): 1.77E E-02 Solvent density (g/m^3) : Solvent visosity (poise): Solution density (g/m^3) at onentraion 1 (g/m^3) : at V Add time on dens eta etarel Y YCAL Z ZCAL Common Interept = Slope (Y) = 7.90E+02 Slope (Z) = -3.44E+02 Intrinsi visosity: m^3/g Huggins onst. k_h: Kraemer onst. k_k: Sum of onstants k_h+k_k: Figure 2. Output data file for VISFIT in the advaned mode, produed by the input data in Figure 1. VISFIT generates also a file with (a) the experimental values of i, y i and z i and (b) two pairs of points, (0,[]) and ( h,y h ), and (0,[]) and ( h,z h ), where h is a value somewhat higher than the highest onentration, and y h and z h are the values alulated for h with the least-squares equations. Eah pair of points defines one or the straight lines Figure 3. The seond output data file for VISFIT, intended for input of a graphis program This seond file (Figure 3) is intended to be imported by graphis program in order to produe a plot of the double-extrapolation. For the data used here as example, we obtain the plot displayed in Figure 4. 6

7 Polystyrene / Toluene / 25 ºC 65 sp / 60 (ln rel )/ sp / (ln rel )/ / g m -3 Figure 3. Plot of the double extrapolation, showing the experimental data and the extrapolating straight lines. The intrinsi visosity of this polystyrene in toluene at 25 ºC was found to be [] = 47.7 m 3 /g. Using the Mark-Houwink-Sakurada equation for this polymer/solvent system, [] = M 0.73, the result for visous-average moleular weight of the polystyrene sample is M = The quality of the results is assured by the exellent agreement of the sum of the Huggins and Kraemer onstants, k H + k K = 0.499, with the exat result in eq Availability The latest version of this program VISFIT program an be found and freely downloaded from our Web site: Exeutable files for MS DOS (running in the ommand mode or under MS Windows), and for Linux will be available, along with sample input and output files in the two operating modes of the program. The names of the exeutables are visfit_x-msd.exe, for MS DOS, and visfit_x-lnx.exe, for Linux, where x is the number of the present version. 6. Release notes This is the first version of the program, submitted along with the final version of the paper in J. Chem. Ed. In future versions this setion will ontain a desription of the modifiations introdued in the most reent version. 7