Hydrodynamic radius of poly (ethylene glycol) 70000 determined under strongly non-ideal solution conditions

Hydrodynamic radius of poly (ethylene glycol) 70000 determined under strongly non-ideal solution conditions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Short Report Hydrodynamic radius of poly (ethylene glycol) 70000 determined under strongly non-ideal solution conditions Adedayo Akinkunmi Fodeke This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6938767/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Hydrodynamic radius of highly concentrated poly (ethylene glycol) 70000 (PEG70K) molecules in phosphate buffer pH 7.4 (total ionic strength between 0.2 – 1.4 mole dm -3 ) was determined using turbidity measurement at 600 nm. Analyses of the distribution equilibrium data of the experiment was carried out by eliminating the effect of high ionic strength on the polymer solution. The maximum of the dependence of natural logarithm of activity coefficient contribution of PEG70K to the phase separation gives the critical concentration (the PEG70K concentration at which the activity coefficient is maximum). The critical concentration was determined by equating the derivative of the quadratic equation of the best-fit curve through the experimental data with respect to concentration to zero. Assumption was made that PEG70K is composed of closely packed, spherical, impenetrable molecules whose energy of attraction is equal to that of repulsion at critical concentration. The hydrodynamic radius value of (9.11±0.32) nm here reported is found to be in good agreement with theoretical estimated value of 8.78 nm for unfolded protein of equivalent molecular weight. This value also lies between 6.59 nm for PEG35000 and 11.95 for PEG100000 experimentally determined from viscosity measurement. PEG70K hydrodynamic radius phase separation activity coefficient equilibrium distribution Figures Figure 1 Figure 2 Introduction The hydrodynamic radius (R h ) of a macromolecule is a crucial parameter for understanding its behavior in solution [ 1 ]. It reflects the molecule's size and shape in relation to the interacting surrounding fluid. Knowledge of R h has found usefulness in various applications in biophysical characterization of molecules [ 2 ], such as understanding their diffusion and aggregation behaviour [ 3 – 4 ] and has been used in the study of protein structure. R h provides information about a molecule's overall behavior. It has been used to complement other structural techniques like X-ray crystallography or NMR and gives insights into how a molecule occupies space in solution as it interacts with the solvent. R h directly impacts the diffusion rate and the apparent molecular weight of macromolecules in solution. It has been demonstrated to play a significant role in physiological processes such as red blood cell aggregation, with smaller molecules (R h 4 nm) promoting it [ 5 ]. R h influences the solution viscosity, particularly in branched macromolecules. Changes in R h can indicate conformational changes during protein folding and unfolding. Comparing the hydrodynamic radius of a protein with that of a globular protein of the same molar mass can help in the classification of its conformation [ 4 ]. R h has been determined and are often determined using techniques like Dynamic Light Scattering (DLS) [ 6 ]. This method requires sample of high purity often difficult to attain, as presence of dust or other contaminants affects measurement accuracy. DLS is not suitable for measurement of high sample concentration because of its sensitivity to large aggregates, and choice of model for analyses can significantly impact on the result. Whereas the use of DLS in measuring the Rh of polymer molecules requires that the solution be sufficiently dilute to avoid inter-particle interactions, viscosity measurement, which is another common method often used comes with the challenge that small error can impact significantly on the accuracy of the result. Here we report a method which imposes no dilute solution condition. Measurements can be made at medium to high concentration of polymer and requires no sophisticated equipment beyond the use of well sample bottles, volumetric flask, thermostated water bath and colorimeter, turbidimeter or visible spectrophotometer. Phosphate buffer pH 7.4 at different ionic strength (I = 0.2–1.4 mol dm − 3 NaCl) was prepared. 0.01–0.05 g PEG70K (Lot # N70620932, product of Nanjing Forever Pharmacy Co. Ltd) with nominal molar mass of 70,000 g/mole was dissolved in phosphate buffer pH 7.4 of given ionic strength (I = 0.2–1.4 mol dm − 3 NaCl) in properly capped sample bottles and maintained at fixed temperature in a Grant thermostated water bath equipped with CC 60 Cryocool immersion cooler, product of Thermoscientific Neslab. This provided temperature stability to within ± 0.1 o C. The bottles were vigorously shaken to dissolve the PEG70K in the buffer solution and then kept in thermostated water bath for four hours to ensure equilibration of the solution of the mixture at different ionic strength. Before taking each sample from each tube into the cuvette for absorbance reading, the equilibrated mixture were mixed thoroughly. The sample was drawn into cuvette for absorbance measurement at 600 nm (A 600 ). The absorbance were measured using a Shimadzu 1800 UV-Vis Spectrophotometer. The process was repeated to obtain a replicate absorbance value subject to 5% error limit. The mean values of absorbance measured at 600 nm were converted to transmittance (T 600 ) and the phase equilibrium distribution (K D ) values of the mixture were calculated using Eq. ( 1 ), $$\:{K}_{D}=\frac{1-{T}_{600\:}}{{T}_{600}}$$ 1 In Eq. ( 1 ) transmittance at 600 nm (T 600 ) is given by 10 − A 600 , which is the absorbance at 600 nm. The dependence of the logarithm of the apparent equilibrium constant of the phase distribution (K D ) on the square root of the ionic strength of the added electrolyte at fixed PEG70K concentration is presented in Fig. 1 . The logarithm of equilibrium constant at various ionic strength were plotted against the square root of the ionic strength according to Debye-Huckel Eq. ( 2 ) [ 7 ]. $$\:logK=\:{logK}^{o}-\:log{{\Gamma\:}}_{1}+A{Z}_{\pm\:2}{I\:}^{1/2}$$ 2 Where K is the apparent equilibrium constant, K o is the thermodynamic equilibrium constant and Γ 1 , A, Z ± and I are the effective activity coefficient of PEG70K, the Debye-Huckel constant, the mean ionic charge and the ionic strength of the solution respectively. Constant A, is dependent on temperature and nature of the solvent. The intercept of the plot of dependence of the apparent equilibrium phase distribution on the square root of ionic strength, at limiting ionic strength (I → 0) gives the logarithm of the equilibrium constant of the phase separation, \(\:\text{l}\text{o}\text{g}\left({K}_{D}^{{\prime\:}}\right)\) of PEG only (in the absence of ions). The plot of the dependence of the logarithm of the equilibrium constant of the hypothetical phase separation in the absence of ions (the intercept of the linear plot described in Fig. 1 ) on the concentration of PEG70K gives a curve that was fitted using second degree polynomial equation described by, $$\:\text{l}\text{o}\text{g}\left({K}_{D}^{{\prime\:}}\right)=\text{log}\left({K}_{D}^{o}\right)+\text{l}\text{o}\text{g}\left({{\Gamma\:}}_{PEG70K}\right)$$ 3 where $$\:\text{l}\text{n}\left({{\Gamma\:}}_{PEG70K}\right)={B}_{ii}{w}_{i}+{B}_{iii}{w}_{i}^{2}$$ 4 In Eq. ( 3 ), \(\:{K}_{D}^{o}\) is the hypothetical thermodynamic equilibrium constant of the phase separation at limiting dilution (independent of concentration) of both PEG70K and ions in the mixture. Γ PEG70K is measure of the effective activity coefficient contribution of PEG70K to the phase distribution. B ii and B iii , are the fitting parameters which are the coefficients of the polynomial equation. Thermodynamically they respectively represent the mean of two and three body interaction parameters of PEG70K. Eq. ( 4 ) quantifies the ease of transfer of additional molecule of PEG70K from liquid phase under dilute condition to the solid phase at arbitrary concentration (increasing the turbidity of the mixture). Curves of Fig. 2 were calculated according to Eq. ( 4 ) using the fitting parameters of the experimental data for the dependence of \(\:\text{l}\text{o}\text{g}\left({K}_{D}^{{\prime\:}}\right)\) on concentration of PEG70K (w PEG70K ) reported in Table 1 , according to Eq. ( 3 ) and Eq. ( 4 ), (figure not shown). Whereas B ii account for coefficient of the attractive interaction under very dilute condition, B iii gives the contribution of the repulsive interaction to ln(Γ PEG70k ) at higher concentration. Table 1 Fitting parameters of the dependence of phase distribution constant of PEG70K at infinite dilution of salt on the concentration of the polymer using Eqs. ( 3 ) and ( 4 ) Fitting parameters Temperature ( o C) 20 25 30 35 40 Log(K’ o ) -1.908 ± 0.038 -1.945 ± 0.113 -2.073 ± 0.061 -2.134 ± 0.057 -2.316 ± 0.148 B ii x 10 3 (g − 1 dm 3 ) 79.2 ± 2.7 88.0 ± 8.2 90.9 ± 4.5 94.9 ± 4.2 110.0 ± 10.8 B iii x 10 3 (g − 2 dm 6 ) -1.04 ± 4.4x10 − 2 -1.20 ± 0.13 -1.2 ± 7.2x10 − 2 -1.3 ± 6.7x10 − 2 -1.6 ± 0.17 Critical conc. g/dm 3 38.14 ± 2.91 36.67 ± 7.51 37.92 ± 4.15 36.50 ± 3.50 34.50 ± 7.27 The maximum occur at a concentration termed “critical concentration” whose value is determined from the differential of the equation of the curve through the experimental data point at each temperature. At maximum, the derivative of the equation through the experimental point with respect to the concentration is equal to zero. Hydrodynamic radius of PEG70K was calculated from the mean critical concentration, treating the PEG70K as hard impenetrable spherical molecule which, at very low concentration (around limiting dilution), are widely separated and do not interact with each other but with the solvent. In this concentration range as the number of molecules per unit volume increases, they attract each other thus leading to linear increase in the logarithm of the activity coefficient with increasing concentration of PEG70K. Later, both attractive and repulsive interaction sums up to give a curve with the maximum as seen in Fig. 2 . As the molecules are brought closer and closer with increasing concentration, a concentration (critical concentration) is reached at which repulsive interaction equals attractive interaction just before repulsive interaction begins to dominate the interaction and the logarithm of the equilibrium constant begins to decrease with increasing concentration of the PEG70K. It should be noted that at critical concentration, the inter-nuclear distance between two PEG70K molecules is equal to 2R h (the distance from the center of one PEG70K molecule to the center of another molecule in mutual contact with it). When the inter-nuclear distance is > > 2R h , no interaction, when inter-nuclear distance is slightly > than 2R h attractive interaction dominate (with some repulsive interaction from slight contact). The critical concentration of PEG70K was obtained by taking the differentiating the equation of each curve with respect to PEG70K concentration and equating it to zero. Taking the nominal molecular weight of PEG70K to be 70000 g/mole, and assuming a hard spherical structure for PEG70K molecule, the hydrodynamic radius was calculated from the mean value of the critical concentration (at the different temperatures). This was done because the value of the critical concentration did not show clear dependence on temperature within experimental error. The calculated mean value of critical concentration was 36.75 ± 1.30 g/dm 3 . Using Avogadro’s number, N A = 6.023×10 23 molecules per mole, the number of molecules of PEG70K in a unit volume (m 3 ) is $$\:\frac{36.75\times\:{10}^{3}\:g/{m}^{3}}{70000\:g/mole}\times\:6.023\times\:{10}^{23}molecules=3.162\times\:{10}^{23}\:molecules\:{m}^{-3}$$ If \(\:v\) is the average volume of each molecule at critical concentration, $$\:v=\:\frac{1}{3.162\times\:{10}^{23}\:molecules\:{m}^{-3}}=3.163\times\:{10}^{-24}\:\:{m}^{3}{\:molecules}^{-1}$$ Since PEG70K is taken to be hard spherical molecule, the hydrodynamic radius, \(\:{R}_{h}\) , of the molecule may be obtained as, $$\:{R}_{h}=\sqrt[3]{\frac{3}{4\pi\:}v}$$ 5 \(\:\pi\:\) is \(\:\frac{22}{7}\) $$\:{R}_{h}=\sqrt[3]{\frac{3}{4\pi\:}3.163\times\:{10}^{-24}\:\:{m}^{3}{\:molecules}^{-1}}=(9.11\pm\:0.32)\:\text{n}m\:$$ In other to gauge the reliability of the calculated R h in this experiment, the nominal molecular weight of PEG70K was imputed into the online program which uses theoretical relationship between hydrodynamic radius of protein and molecular weight to estimate the R h . This fluidic science converter for determination of hydrodynamic radius is available at, https://fluidic.com/molecular-weight-to-hydrodynamic-radius-converter/ . The value of 8.78 nm for the hydrodynamic radius of unfolded protein of molar mass of 70,000 is in good agreement with the value of 9.11 ± 0.32 nm here reported [ 8 ]. Also, the value of the hydrodynamic radius obtained by is in good agreement with what would is expected given that viscosity measurement using molecular interaction from distribution equilibrium gave 6.59 nm for PEG35000 and 11.95 for PEG100000 [ 5 ]. The value 9.11 ± 0.32 nm of our method, within experimental error, lies within expected value. The new method developed here which is easy to handle and requires less sophisticated equipment than those conventionally used for the determination of R h , can be useful for its estimation particularly in environment where very expensive equipment like analytical centrifuge and light scattering equipment are unavailable. The method described here is less prone to measurement error unlike R h measurement using viscometer. In an earlier report [ 9 ], involving quantifying attractive interaction between dextran T70 and superoxide dismutase under sedimentation equilibrium condition, the critical concentration was shown to depend on temperature unlike what was observed here. We posit that equilibrium distribution technique if investigated further, may also be useful in studying other hetero molecular interactions. Declarations Funding Declaration No funding has been made available for this work. Data Availability Declaration The complete data in respect of this manuscript are available on request. Conflict of Interest There is no competing interest, be it commercial or otherwise in relation to the work here reported. References Ye, X. Yanga, J, Ambreena J. Scaling laws between the hydrodynamic parameters and molecular weight of linear poly(2-ethyl-2- oxazoline) RSC Adv. 2013; 3:15108–15113. Lee H, Venable RM, MacKerell AD, Pastor RW. Molecular Dynamics Studies of Polyethylene Oxide and Polyethylene Glycol: Hydrodynamic Radius and Shape Anisotropy Biophys. J. 2008;95:1590–1599.. Turetta, L., and Lattuada, M., The role of hydrodynamic interactions on the aggregation kinetics of sedimenting colloidal particles. Soft Matter. 2022; 18 : 1715-1730. Akbarzadehlaleh P, Mirzaei M, Mashahdi-Keshtiban, M, Heidari HR. The Effect of Length and Structure of Attached Polyethylene Glycol Chain on Hydrodynamic Radius, and Separation of PEGylated Human Serum Albumin by Chromatography. Adv Pharm Bull. 2020;11:728–738. Armstrong JK, Wenby RB, Meiselman HJ, Fisher TC. The Hydrodynamic Radii of Macromolecules and Their Effect on Red Blood Cell Aggregation. Biophys. J. 2004;87:4259–4270 Linegar KL, Adeniran AE, Kostko AF, Anisimov MA. Hydrodynamic Radius of Polyethylene Glycol in Solution Obtained by Dynamic Light Scattering. Colloid Journal, 2010;72:279–281. Debye P, Hückel E. Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen (the theory of electrolytes. I. Lowering of freezing point and related phenomena), Physikalische Z. 1923;24:185-206. Fleming J, Fleming KG. Fast Calculations of Folded and Disordered Protein and Nucleic Acid Hydrodynamic Properties. Biophys J., 2018;114:856-869. Fodeke AA, Quantitative characterization of temperature‑independent polymer–polymer interaction and temperature‑dependent protein–protein and protein–polymer interactions in concentrated polymer solutions. Eur Biophys J. 2019;48:189–202. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6938767","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Short Report","associatedPublications":[],"authors":[{"id":489431675,"identity":"0a0558c0-4975-48cf-b65b-c0e4c27eb0b0","order_by":0,"name":"Adedayo Akinkunmi Fodeke","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2ElEQVRIiWNgGAWjYDAC5gMMEgwHGOQYGHhgQmwEtLAlgLUYk64lsYFoLfxtzAdvfDhjk77h+NmDDz4w2MkxSKQl4NUicYwt2XLGjbTcDWfykg1nMCQbA7UcwG/N/R4zaZ4Ph3M3HMgBMhgOJDZIpDfg1SF/jMdM+s+H/+kG598QqcUApIXhxoEEgxtwWwg4zBDkl54zyYYzb7wxNpxhkGzMxvMsAa8WuWPAEPtxzE6e73yO4YMPFXZy/OxpBni1wIEC2DUGhCMSAeQbiFY6CkbBKBgFIw0AAK1qRxryKZ9qAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-9524-8097","institution":"Obafemi Awolowo University","correspondingAuthor":true,"prefix":"","firstName":"Adedayo","middleName":"Akinkunmi","lastName":"Fodeke","suffix":""}],"badges":[],"createdAt":"2025-06-20 12:09:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6938767/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6938767/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":87562310,"identity":"99bf5902-1ce5-46a7-aaa1-51ec01bce8b2","added_by":"auto","created_at":"2025-07-25 08:45:54","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":69993,"visible":true,"origin":"","legend":"\u003cp\u003eDependence of the logarithm of the apparent equilibrium constant of phase separation of PEG70K on the square root of the ionic strength in phosphate buffer pH 7.4 (ionic strength adjusted with NaCl) for (a) 10 g/dm\u003csup\u003e3\u003c/sup\u003e PEG70K (b) 50 g/dm\u003csup\u003e3\u003c/sup\u003e PEG70K. Symbols and line:\u0026nbsp; 20\u003csup\u003eo\u003c/sup\u003eC (circle and solid line) and 35\u003csup\u003eo\u003c/sup\u003eC (square and broken line).\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6938767/v1/1508022839caeb9646964b6f.png"},{"id":87562309,"identity":"1858cae9-eb61-4a64-adb0-70853cd6c3b8","added_by":"auto","created_at":"2025-07-25 08:45:54","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":77587,"visible":true,"origin":"","legend":"\u003cp\u003eDependence of the natural logarithm of the net activity coefficient of the PEG70k in the two phases (at infinite dilution of ions) on the concentration of the PEG70K at 20\u003csup\u003eo\u003c/sup\u003eC (solid curve), 25\u003csup\u003eo\u003c/sup\u003eC (long broken curve), 30\u003csup\u003eo\u003c/sup\u003eC (short broken curve), 35\u003csup\u003eo\u003c/sup\u003eC (short dotted broken curve) and 40\u003csup\u003eo\u003c/sup\u003eC (grid dotted curve). Curves were fitted with Eq. (4) using the fitting parameters of Table 1; rows 3 and 4 (columns 2 – 6).\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6938767/v1/785a8f52a35b11cf9473e0cf.png"},{"id":91777423,"identity":"d4873bfe-338d-4544-8e0a-cc2b312f73a8","added_by":"auto","created_at":"2025-09-20 21:38:40","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":451233,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6938767/v1/9d9cedb2-1814-4886-b9fe-b48eeb511699.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003eHydrodynamic radius of poly (ethylene glycol) 70000 determined under strongly non-ideal solution conditions\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe hydrodynamic radius (R\u003csub\u003eh\u003c/sub\u003e) of a macromolecule is a crucial parameter for understanding its behavior in solution [\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e]. It reflects the molecule\u0026apos;s size and shape in relation to the interacting surrounding fluid. Knowledge of R\u003csub\u003eh\u003c/sub\u003e has found usefulness in various applications in biophysical characterization of molecules [\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e], such as understanding their diffusion and aggregation behaviour [\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e] and has been used in the study of protein structure.\u003c/p\u003e\n\u003cp\u003eR\u003csub\u003eh\u003c/sub\u003e provides information about a molecule\u0026apos;s overall behavior. It has been used to complement other structural techniques like X-ray crystallography or NMR and gives insights into how a molecule occupies space in solution as it interacts with the solvent. R\u003csub\u003eh\u003c/sub\u003e directly impacts the diffusion rate and the apparent molecular weight of macromolecules in solution. It has been demonstrated to play a significant role in physiological processes such as red blood cell aggregation, with smaller molecules (R\u003csub\u003eh\u003c/sub\u003e \u0026lt; 4 nm) inhibiting aggregation and larger ones (Rh\u0026thinsp;\u0026gt;\u0026thinsp;4 nm) promoting it [\u003cspan class=\"CitationRef\"\u003e5\u003c/span\u003e]. R\u003csub\u003eh\u003c/sub\u003e influences the solution viscosity, particularly in branched macromolecules. Changes in R\u003csub\u003eh\u003c/sub\u003e can indicate conformational changes during protein folding and unfolding. Comparing the hydrodynamic radius of a protein with that of a globular protein of the same molar mass can help in the classification of its conformation [\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e]. R\u003csub\u003eh\u003c/sub\u003e has been determined and are often determined using techniques like Dynamic Light Scattering (DLS) [\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e]. This method requires sample of high purity often difficult to attain, as presence of dust or other contaminants affects measurement accuracy. DLS is not suitable for measurement of high sample concentration because of its sensitivity to large aggregates, and choice of model for analyses can significantly impact on the result. Whereas the use of DLS in measuring the Rh of polymer molecules requires that the solution be sufficiently dilute to avoid inter-particle interactions, viscosity measurement, which is another common method often used comes with the challenge that small error can impact significantly on the accuracy of the result. Here we report a method which imposes no dilute solution condition. Measurements can be made at medium to high concentration of polymer and requires no sophisticated equipment beyond the use of well sample bottles, volumetric flask, thermostated water bath and colorimeter, turbidimeter or visible spectrophotometer.\u003c/p\u003e\n\u003cp\u003ePhosphate buffer pH 7.4 at different ionic strength (I\u0026thinsp;=\u0026thinsp;0.2\u0026ndash;1.4 mol dm\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e NaCl) was prepared. 0.01\u0026ndash;0.05 g PEG70K (Lot # N70620932, product of Nanjing Forever Pharmacy Co. Ltd) with nominal molar mass of 70,000 g/mole was dissolved in phosphate buffer pH 7.4 of given ionic strength (I\u0026thinsp;=\u0026thinsp;0.2\u0026ndash;1.4 mol dm\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e NaCl) in properly capped sample bottles and maintained at fixed temperature in a Grant thermostated water bath equipped with CC 60 Cryocool immersion cooler, product of Thermoscientific Neslab. This provided temperature stability to within \u0026plusmn;\u0026thinsp;0.1\u003csup\u003eo\u003c/sup\u003eC. The bottles were vigorously shaken to dissolve the PEG70K in the buffer solution and then kept in thermostated water bath for four hours to ensure equilibration of the solution of the mixture at different ionic strength. Before taking each sample from each tube into the cuvette for absorbance reading, the equilibrated mixture were mixed thoroughly. The sample was drawn into cuvette for absorbance measurement at 600 nm (A\u003csub\u003e600\u003c/sub\u003e). The absorbance were measured using a Shimadzu 1800 UV-Vis Spectrophotometer. The process was repeated to obtain a replicate absorbance value subject to 5% error limit. The mean values of absorbance measured at 600 nm were converted to transmittance (T\u003csub\u003e600\u003c/sub\u003e) and the phase equilibrium distribution (K\u003csub\u003eD\u003c/sub\u003e) values of the mixture were calculated using Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e),\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\:{K}_{D}=\\frac{1-{T}_{600\\:}}{{T}_{600}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e) transmittance at 600 nm (T\u003csub\u003e600\u003c/sub\u003e) is given by 10\u003csup\u003e\u0026minus;\u0026thinsp;A\u003c/sup\u003e\u003csub\u003e600\u003c/sub\u003e, which is the absorbance at 600 nm.\u003c/p\u003e\n\u003cp\u003eThe dependence of the logarithm of the apparent equilibrium constant of the phase distribution (K\u003csub\u003eD\u003c/sub\u003e) on the square root of the ionic strength of the added electrolyte at fixed PEG70K concentration is presented in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThe logarithm of equilibrium constant at various ionic strength were plotted against the square root of the ionic strength according to Debye-Huckel Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e) [\u003cspan class=\"CitationRef\"\u003e7\u003c/span\u003e].\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$$\\:logK=\\:{logK}^{o}-\\:log{{\\Gamma\\:}}_{1}+A{Z}_{\\pm\\:2}{I\\:}^{1/2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere K is the apparent equilibrium constant, K\u003csup\u003eo\u003c/sup\u003e is the thermodynamic equilibrium constant and \u0026Gamma;\u003csub\u003e1\u003c/sub\u003e, A, Z\u003csub\u003e\u0026plusmn;\u003c/sub\u003e and I are the effective activity coefficient of PEG70K, the Debye-Huckel constant, the mean ionic charge and the ionic strength of the solution respectively. Constant A, is dependent on temperature and nature of the solvent. The intercept of the plot of dependence of the apparent equilibrium phase distribution on the square root of ionic strength, at limiting ionic strength (I \u0026rarr; 0) gives the logarithm of the equilibrium constant of the phase separation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{l}\\text{o}\\text{g}\\left({K}_{D}^{{\\prime\\:}}\\right)\\)\u003c/span\u003e\u003c/span\u003e of PEG only (in the absence of ions). The plot of the dependence of the logarithm of the equilibrium constant of the hypothetical phase separation in the absence of ions (the intercept of the linear plot described in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e) on the concentration of PEG70K gives a curve that was fitted using second degree polynomial equation described by,\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$$\\:\\text{l}\\text{o}\\text{g}\\left({K}_{D}^{{\\prime\\:}}\\right)=\\text{log}\\left({K}_{D}^{o}\\right)+\\text{l}\\text{o}\\text{g}\\left({{\\Gamma\\:}}_{PEG70K}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere\u003c/p\u003e\n\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e$$\\:\\text{l}\\text{n}\\left({{\\Gamma\\:}}_{PEG70K}\\right)={B}_{ii}{w}_{i}+{B}_{iii}{w}_{i}^{2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{K}_{D}^{o}\\)\u003c/span\u003e\u003c/span\u003e is the hypothetical thermodynamic equilibrium constant of the phase separation at limiting dilution (independent of concentration) of both PEG70K and ions in the mixture. \u0026Gamma;\u003csub\u003ePEG70K\u003c/sub\u003e is measure of the effective activity coefficient contribution of PEG70K to the phase distribution. B\u003csub\u003eii\u003c/sub\u003e and B\u003csub\u003eiii\u003c/sub\u003e, are the fitting parameters which are the coefficients of the polynomial equation. Thermodynamically they respectively represent the mean of two and three body interaction parameters of PEG70K. Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e) quantifies the ease of transfer of additional molecule of PEG70K from liquid phase under dilute condition to the solid phase at arbitrary concentration (increasing the turbidity of the mixture). Curves of Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e were calculated according to Eq. (\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e) using the fitting parameters of the experimental data for the dependence of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{l}\\text{o}\\text{g}\\left({K}_{D}^{{\\prime\\:}}\\right)\\)\u003c/span\u003e\u003c/span\u003e on concentration of PEG70K (w\u003csub\u003ePEG70K\u003c/sub\u003e) reported in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, according to Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e) and Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e), (figure not shown). Whereas B\u003csub\u003eii\u003c/sub\u003e account for coefficient of the attractive interaction under very dilute condition, B\u003csub\u003eiii\u003c/sub\u003e gives the contribution of the repulsive interaction to ln(\u0026Gamma;\u003csub\u003ePEG70k\u003c/sub\u003e) at higher concentration.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eFitting parameters of the dependence of phase distribution constant of PEG70K at infinite dilution of salt on the concentration of the polymer using Eqs.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e) and (\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e)\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFitting parameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"5\"\u003e\n \u003cp\u003eTemperature (\u003csup\u003eo\u003c/sup\u003eC)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e40\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eLog(K\u0026rsquo;\u003csup\u003eo\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.908\u0026thinsp;\u0026plusmn;\u0026thinsp;0.038\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.945\u0026thinsp;\u0026plusmn;\u0026thinsp;0.113\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-2.073\u0026thinsp;\u0026plusmn;\u0026thinsp;0.061\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-2.134\u0026thinsp;\u0026plusmn;\u0026thinsp;0.057\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-2.316\u0026thinsp;\u0026plusmn;\u0026thinsp;0.148\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eB\u003csub\u003eii\u003c/sub\u003e x 10\u003csup\u003e3\u003c/sup\u003e (g\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003edm\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e79.2\u0026thinsp;\u0026plusmn;\u0026thinsp;2.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e88.0\u0026thinsp;\u0026plusmn;\u0026thinsp;8.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e90.9\u0026thinsp;\u0026plusmn;\u0026thinsp;4.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e94.9\u0026thinsp;\u0026plusmn;\u0026thinsp;4.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e110.0\u0026thinsp;\u0026plusmn;\u0026thinsp;10.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eB\u003csub\u003eiii\u003c/sub\u003e x 10\u003csup\u003e3\u003c/sup\u003e (g\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003edm\u003csup\u003e6\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.04\u0026thinsp;\u0026plusmn;\u0026thinsp;4.4x10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.20\u0026thinsp;\u0026plusmn;\u0026thinsp;0.13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.2\u0026thinsp;\u0026plusmn;\u0026thinsp;7.2x10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.3\u0026thinsp;\u0026plusmn;\u0026thinsp;6.7x10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.6\u0026thinsp;\u0026plusmn;\u0026thinsp;0.17\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCritical conc. g/dm\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e38.14\u0026thinsp;\u0026plusmn;\u0026thinsp;2.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e36.67\u0026thinsp;\u0026plusmn;\u0026thinsp;7.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e37.92\u0026thinsp;\u0026plusmn;\u0026thinsp;4.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e36.50\u0026thinsp;\u0026plusmn;\u0026thinsp;3.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e34.50\u0026thinsp;\u0026plusmn;\u0026thinsp;7.27\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003eThe maximum occur at a concentration termed \u0026ldquo;critical concentration\u0026rdquo; whose value is determined from the differential of the equation of the curve through the experimental data point at each temperature. At maximum, the derivative of the equation through the experimental point with respect to the concentration is equal to zero. Hydrodynamic radius of PEG70K was calculated from the mean critical concentration, treating the PEG70K as hard impenetrable spherical molecule which, at very low concentration (around limiting dilution), are widely separated and do not interact with each other but with the solvent. In this concentration range as the number of molecules per unit volume increases, they attract each other thus leading to linear increase in the logarithm of the activity coefficient with increasing concentration of PEG70K. Later, both attractive and repulsive interaction sums up to give a curve with the maximum as seen in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. As the molecules are brought closer and closer with increasing concentration, a concentration (critical concentration) is reached at which repulsive interaction equals attractive interaction just before repulsive interaction begins to dominate the interaction and the logarithm of the equilibrium constant begins to decrease with increasing concentration of the PEG70K. It should be noted that at critical concentration, the inter-nuclear distance between two PEG70K molecules is equal to 2R\u003csub\u003eh\u003c/sub\u003e (the distance from the center of one PEG70K molecule to the center of another molecule in mutual contact with it). When the inter-nuclear distance is \u0026gt;\u0026thinsp;\u0026gt;\u0026thinsp;2R\u003csub\u003eh\u003c/sub\u003e, no interaction, when inter-nuclear distance is slightly\u0026thinsp;\u0026gt;\u0026thinsp;than 2R\u003csub\u003eh\u003c/sub\u003e attractive interaction dominate (with some repulsive interaction from slight contact). The critical concentration of PEG70K was obtained by taking the differentiating the equation of each curve with respect to PEG70K concentration and equating it to zero. Taking the nominal molecular weight of PEG70K to be 70000 g/mole, and assuming a hard spherical structure for PEG70K molecule, the hydrodynamic radius was calculated from the mean value of the critical concentration (at the different temperatures). This was done because the value of the critical concentration did not show clear dependence on temperature within experimental error. The calculated mean value of critical concentration was 36.75\u0026thinsp;\u0026plusmn;\u0026thinsp;1.30 g/dm\u003csup\u003e3\u003c/sup\u003e. Using Avogadro\u0026rsquo;s number, N\u003csub\u003eA\u003c/sub\u003e = 6.023\u0026times;10\u003csup\u003e23\u003c/sup\u003e molecules per mole, the number of molecules of PEG70K in a unit volume (m\u003csup\u003e3\u003c/sup\u003e) is\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$$\\:\\frac{36.75\\times\\:{10}^{3}\\:g/{m}^{3}}{70000\\:g/mole}\\times\\:6.023\\times\\:{10}^{23}molecules=3.162\\times\\:{10}^{23}\\:molecules\\:{m}^{-3}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIf \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v\\)\u003c/span\u003e\u003c/span\u003e is the average volume of each molecule at critical concentration,\u003c/p\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e$$\\:v=\\:\\frac{1}{3.162\\times\\:{10}^{23}\\:molecules\\:{m}^{-3}}=3.163\\times\\:{10}^{-24}\\:\\:{m}^{3}{\\:molecules}^{-1}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSince PEG70K is taken to be hard spherical molecule, the hydrodynamic radius, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{R}_{h}\\)\u003c/span\u003e\u003c/span\u003e, of the molecule may be obtained as,\u003c/p\u003e\n\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e$$\\:{R}_{h}=\\sqrt[3]{\\frac{3}{4\\pi\\:}v}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\pi\\:\\)\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{22}{7}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e$$\\:{R}_{h}=\\sqrt[3]{\\frac{3}{4\\pi\\:}3.163\\times\\:{10}^{-24}\\:\\:{m}^{3}{\\:molecules}^{-1}}=(9.11\\pm\\:0.32)\\:\\text{n}m\\:$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn other to gauge the reliability of the calculated R\u003csub\u003eh\u003c/sub\u003e in this experiment, the nominal molecular weight of PEG70K was imputed into the online program which uses theoretical relationship between hydrodynamic radius of protein and molecular weight to estimate the R\u003csub\u003eh\u003c/sub\u003e. This fluidic science converter for determination of hydrodynamic radius is available at, \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://fluidic.com/molecular-weight-to-hydrodynamic-radius-converter/\u003c/span\u003e\u003c/span\u003e. The value of 8.78 nm for the hydrodynamic radius of unfolded protein of molar mass of 70,000 is in good agreement with the value of 9.11\u0026thinsp;\u0026plusmn;\u0026thinsp;0.32 nm here reported [\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e]. Also, the value of the hydrodynamic radius obtained by is in good agreement with what would is expected given that viscosity measurement using molecular interaction from distribution equilibrium gave 6.59 nm for PEG35000 and 11.95 for PEG100000 [\u003cspan class=\"CitationRef\"\u003e5\u003c/span\u003e]. The value 9.11\u0026thinsp;\u0026plusmn;\u0026thinsp;0.32 nm of our method, within experimental error, lies within expected value. The new method developed here which is easy to handle and requires less sophisticated equipment than those conventionally used for the determination of R\u003csub\u003eh\u003c/sub\u003e, can be useful for its estimation particularly in environment where very expensive equipment like analytical centrifuge and light scattering equipment are unavailable. The method described here is less prone to measurement error unlike R\u003csub\u003eh\u003c/sub\u003e measurement using viscometer. In an earlier report [\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e], involving quantifying attractive interaction between dextran T70 and superoxide dismutase under sedimentation equilibrium condition, the critical concentration was shown to depend on temperature unlike what was observed here. We posit that equilibrium distribution technique if investigated further, may also be useful in studying other hetero molecular interactions.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding Declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo funding has been made available for this work.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability Declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe complete data in respect of this manuscript are available on request.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThere is no competing interest, be it commercial or otherwise in relation to the work here reported.\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eYe, X. Yanga, J, Ambreena J. Scaling laws between the hydrodynamic parameters and molecular weight of linear poly(2-ethyl-2- oxazoline) RSC Adv. 2013; 3:15108\u0026ndash;15113.\u003c/li\u003e\n\u003cli\u003eLee H, Venable RM, MacKerell AD, Pastor RW. Molecular Dynamics Studies of Polyethylene Oxide and Polyethylene Glycol: Hydrodynamic Radius and Shape Anisotropy Biophys. J. 2008;95:1590\u0026ndash;1599..\u003c/li\u003e\n\u003cli\u003eTuretta, L., and Lattuada, M., The role of hydrodynamic interactions on the aggregation kinetics of sedimenting colloidal particles. Soft Matter. 2022;\u003cstrong\u003e18\u003c/strong\u003e: 1715-1730. \u003c/li\u003e\n\u003cli\u003eAkbarzadehlaleh P, Mirzaei M, Mashahdi-Keshtiban, M, Heidari HR. The Effect of Length and Structure of Attached Polyethylene Glycol Chain on Hydrodynamic Radius, and Separation of PEGylated Human Serum Albumin by Chromatography. Adv Pharm Bull. 2020;11:728\u0026ndash;738.\u003c/li\u003e\n\u003cli\u003eArmstrong JK, Wenby RB, Meiselman HJ, Fisher TC. The Hydrodynamic Radii of Macromolecules and Their Effect on Red Blood Cell Aggregation. Biophys. J. 2004;87:4259\u0026ndash;4270\u003c/li\u003e\n\u003cli\u003eLinegar KL, Adeniran AE, Kostko AF, Anisimov MA. Hydrodynamic Radius of Polyethylene Glycol in Solution Obtained by Dynamic Light Scattering. Colloid Journal, 2010;72:279\u0026ndash;281.\u003c/li\u003e\n\u003cli\u003eDebye P, H\u0026uuml;ckel E. Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen (the theory of electrolytes. I. Lowering of freezing point and related phenomena), Physikalische Z. 1923;24:185-206.\u003c/li\u003e\n\u003cli\u003eFleming J, Fleming KG. Fast Calculations of Folded and Disordered Protein and Nucleic Acid Hydrodynamic Properties. Biophys J., 2018;114:856-869.\u003c/li\u003e\n\u003cli\u003eFodeke AA, Quantitative characterization of temperature‑independent polymer\u0026ndash;polymer interaction and temperature‑dependent protein\u0026ndash;protein and protein\u0026ndash;polymer interactions in concentrated polymer solutions. Eur Biophys J. 2019;48:189\u0026ndash;202.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"PEG70K, hydrodynamic radius, phase separation, activity coefficient, equilibrium distribution","lastPublishedDoi":"10.21203/rs.3.rs-6938767/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6938767/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHydrodynamic radius of highly concentrated poly (ethylene glycol) 70000 (PEG70K) molecules in phosphate buffer pH 7.4 (total ionic strength between 0.2 – 1.4 mole dm\u003csup\u003e-3\u003c/sup\u003e) was determined using turbidity measurement at 600 nm. Analyses of the distribution equilibrium data of the experiment was carried out by eliminating the effect of high ionic strength on the polymer solution. The maximum of the dependence of natural logarithm of activity coefficient contribution of PEG70K to the phase separation gives the critical concentration (the PEG70K concentration at which the activity coefficient is maximum). The critical concentration was determined by equating the derivative of the quadratic equation of the best-fit curve through the experimental data with respect to concentration to zero. Assumption was made that PEG70K is composed of closely packed, spherical, impenetrable molecules whose energy of attraction is equal to that of repulsion at critical concentration. The hydrodynamic radius value of (9.11±0.32) nm here reported is found to be in good agreement with theoretical estimated value of 8.78 nm for unfolded protein of equivalent molecular weight. \u0026nbsp;This value also lies between 6.59 nm for PEG35000 and 11.95 for PEG100000 experimentally determined from viscosity measurement.\u003c/p\u003e","manuscriptTitle":"Hydrodynamic radius of poly (ethylene glycol) 70000 determined under strongly non-ideal solution conditions","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-25 08:45:49","doi":"10.21203/rs.3.rs-6938767/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"197816a4-e4b1-439e-85f4-45044736b3b7","owner":[],"postedDate":"July 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-09-20T21:30:33+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-25 08:45:49","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6938767","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6938767","identity":"rs-6938767","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}