Quantification of Polymer-Surface Interaction using Microcalorimetry

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Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Quantification of Polymer−Surface Interaction Using Microcalorimetry Lalaso V. Mohite,*,† Vinay A. Juvekar,‡ and Jyoti Sahu‡ †

Aditya Birla Science and Technology Company Private Limited, Navi Mumbai, Maharashtra 410208, India Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra 400076, India

‡

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S Supporting Information *

ABSTRACT: Isothermal titration microcalorimetry experiments are performed to measure the heat of adsorption of poly(ethylene glycol) (PEG) on nonporous silica nanoparticles. A calorimetric technique is also developed for the determination of the adsorption isotherm for the PEG−silica system. A model based on the continuum form of the selfconsistent field theory is used for estimating the polymer− surface interaction parameter, χ*, from the heat of adsorption of the polymer. The effect of temperature and molecular weight of PEG on χ* has been determined. It is shown that the adsorbed amount of PEG on silica, predicted using the estimated value of χ*, is in good agreement with the experimental data. We also find that the major contribution to the heat of adsorption arises from the bound fraction of the adsorbed polymer. model has been used for evaluation of χ* from the critical displacer concentration.11 The values of the interaction parameter for several polymer−solvent−solid systems have been estimated using this method.12−18 The main drawback of the adsorption/desorption-transition method is that it requires the use of a third component (displacer). The applicability of the method hinges on the assumption that χ* is independent of the nature of the displacer used. Although van der Beek et al.14 have shown that χ* is insensitive, within experimental errors, to the nature of displacer used, their conclusion is based on limited experimental data and cannot be generalized. In principle, the interactions of the displacer with the solvent, the surface, and the polymer cannot be completely ignored and a certain element of doubt is always associated with the estimates of χ* by this method. Therefore, need exists for a method which avoids the use of displacer and estimates the absolute value of χ*. One possible method is the measurement of the heat of adsorption of polymer from solvent to the solid surface. A calorimetric technique could be used for this purpose. Although this technique has been used for measurement of bound fractions of the polymer on the surface,20−23 it has not been investigated sufficiently well to prove its utility for the measurement of χ*. Cohen Stuart et al.20

1. INTRODUCTION Polymer adsorption is used as a technique to manipulate properties of solid−liquid interfaces for applications such as colloid stabilization,1,2 flocculation,1,2 modification of surface wettability,3 lubrication,4,5 drag reduction,6,7 and oil recovery,8,9 etc. The affinity of the polymer toward the surface plays an important role in its selection for these applications. For example, in the case of colloid stabilization, the polymer should have a high affinity for both the surface as well as the solvent in order to produce a thick and well extended adsorbed layer having a strong steric repulsive energy barrier. The strength of the polymer−surface interaction is quantified by the interaction parameter χ*. According to Silberberg,10 this parameter is defined as the segmental adsorption energy (in units of kT) of the polymer relative to that of the solvent molecule. Adsorption of the polymer results only if χ* exceeds a critical value χ*c . Hence a polymer which adsorbs on a solid in one solvent may not do so in some other solvent. For selecting a suitable solvent for adsorption and also for predicting the extent of adsorption, it is necessary to know the χ* parameter accurately for the given polymer−solvent−solid system. The most widely used method for estimating χ* involves desorption of the polymer by a displacer. In this method, the polymer is adsorbed on the solid surface from its solution in a given solvent. Then, desorption of the polymer from the surface is performed by introducing a displacer, which is more strongly adsorbing than the polymer. At the critical concentration of the displacer, the polymer is fully desorbed from the surface. A variety of techniques have been employed to estimate the critical displacer concentration.11−19 The analytical form of the lattice © XXXX American Chemical Society

Special Issue: Vinay Juvekar Festschrift Received: Revised: Accepted: Published: A

October 2, 2018 December 21, 2018 December 25, 2018 December 25, 2018 DOI: 10.1021/acs.iecr.8b04792 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

by the manufacturer. This indicates that silica nanoparticles were monodisperse and have low surface roughness.27 Two types of experiments were performed. In the first, we measured the heat of adsorption of PEG on silica in order to estimate the polymer−surface interaction parameter. In the second, we measured the adsorption of PEG on silica using the depletion method, in which microcalorimetry was used to estimate the concentration of PEG in the depleted solution. 2.1. Measurement of the Heat of Adsorption. The heat of adsorption of PEG on the silica surface was measured by isothermal titration calorimeter CSC 4200 (Calorimetry Science Corporation, USA). The sensitivity of the calorimeter is ±0.5 μJ. The calorimeter was precalibrated using a precise electrical input. The calorimetric cell was a cylindrical stainless steel vessel having 1.3 mL volume. A turbine agitator (2-bladed, 5 mm diameter), provided with the cell, was set at 300 rpm to give uniform stirring to the solution in the cell. The thermoelectric device of the instrument detected the heat evolved during the adsorption of the polymer. Initially, the cell was completely filled with 1.3 mL of the silica suspension. The polymer solution was then injected into the cell from a gastight syringe, through a stainless steel cannula. A computer-controlled syringe pump of the instrument was used for the injection. The injected liquid displaced the corresponding volume of the fluid from the cell into the annular space between the needle and the cannula. An appropriate correction was incorporated to account for the enthalpy content of the displaced liquid. The silica nanoparticle suspension was prepared by diluting the concentrated suspension (34% w/w silica) with deionized water, whereas PEG solution was prepared by mixing the solid polymer with deionized water. Concentrations of the silica nanoparticle and the polymer in the cell and the syringe were selected in such a way that the quantity of the polymer added per injection covered only a small fraction of the surface area of the particles, but at the same time, the heat evolved per injection was much larger than the least count of the calorimeter (0.5 μJ). The interval between two consecutive injections was kept in the range of 500−2000 s. This was found to be adequate for the complete evolution of the heat as indicated by the return of the thermogram to the baseline after the peak. The concentration of silica nanoparticles in the cell was 0.02 (w/w) in all experiments. The concentration of the polymer solution in the syringe ranged from 0.15 to 10 g·L−1. The experiments were performed at three temperatures, namely, 298.15 K, 308.15 K, and 318.15 K. Either 4 μL or 8 μL of the polymer solution was used per injection. In a few experiments, reverse titration was performed, in which a PEG solution in the cell was titrated against a suspension of silica nanoparticles in the syringe. Figure 1 shows a typical thermogram from the experiment on isothermal titration of silica nanoparticle suspension in the cell with PEG100K solution in the syringe. The ordinate represents the rate of evolution of heat. The area under the peak represents the heat evolved during the injection and it is denoted by −Q1j, for jth injection. The heat evolved during the titration consists of two parts: the heat of dilution of PEG by water in the cell and the heat of adsorption of PEG on silica. To estimate the heat of dilution, the supernatant solution (obtained by centrifugation of the silica suspension), was placed in the cell and titrated against the PEG100K solution in the syringe. Figure 2 shows the thermogram obtained. The area under the peak represents the

pointed out that the enthalpy data are not suitable for the measurement of the train fraction of the polymer (fraction of the chain segments in direct contact with the solid) for high molecular weight polymers because the heat of adsorption evolves very slowly and hence is always underestimated. However, the microcalorimeter used by Cohen Stuart et al.20 did not have adequate sensitivity. In 1993, Trens and Denoyel23 used a high sensitivity microcalorimeter for the measurement of heat of adsorption of poly(ethylene oxide) on silica. They demonstrated the applicability of microcalorimetry for the measurement of the train fraction even for high molecular weight polymers. The present work is undertaken to test the suitability of the calorimetric technique for the measurement of χ*. The heat of adsorption of poly(ethylene glycols) (PEG) (molecular weights ranging from 8 kDa to 100 kDa) on silica nanoparticles (Ludox TMA) have been measured at different temperatures using an isothermal titration calorimeter. The particles used here are monodisperse, spherical, and nonporous. Moreover, they do not agglomerate and hence do not require the additive for stabilization. We also observed that the particles do not flocculate during the process of adsorption of PEG. This allowed us to assume that the entire particle surface area is accessible to the polymer for adsorption. We have developed a model based on the continuum form of the self-consistent field theory for an estimation of the χ* parameter from the integral enthalpy of adsorption. χ* is found to be weakly dependent on the temperature and molecular weight of the polymer. In addition to measurement of χ*, we have also used isothermal titration calorimetry to estimate the adsorption isotherm of PEG on silica. The work is presented as follows. First, the experimental procedure used for the measurement of the heat of adsorption is described. Then the use of this procedure to estimate the adsorption isotherm of PEG on silica is discussed. This is followed by the description of the model for the estimation of χ* from the integral enthalpy of adsorption. The procedure for regression of the experimental data to determine χ* is described next. The accuracy and sensitivity of the estimated values of χ* are then discussed. The discussion also includes the analysis of the calorimetric data of Trens and Denoyel23 and analysis of the hypothesis of Killman et al.22

2. EXPERIMENTAL SECTION PEG having molecular weights of 8 kDa (PEG8K), 20 kDa (PEG20K), and 100 kDa (PEG100K) were purchased from Sigma-Aldrich. The following values of the polydispersity indices (PDI) for different molecular weights of PEG, supplied by Sigma-Aldrich, have been reported in the literature, based on GPC analysis: 1.05 for PEG8K,24 1.08 for PEG20K,24 and 1.12 for PEG100K.25 These PDI values show that all PEG samples used were fairly monodisperse. Hence they were used as supplied. The self-stabilized silica nanoparticles (Ludox TMA) having the surface area of 140 m2·g−1 were obtained from SigmaAldrich as 0.34 (w/w) suspension in deionized water. This suspension has pH 7 and particles are negatively charged (zeta potential, −60 mV).26 The diameter of these silica nanoparticles, measured using dynamic light scattering was 20.1 ± 0.4 nm. The geometric surface area calculated from this particle diameter and using the density of silica as 2.2 g·cm−3, was 136 m2·g−1, which was close to the specific surface area of 140 m2·g−1, as provided B

DOI: 10.1021/acs.iecr.8b04792 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research ΔQ̂ D = j

Q 1j − Q 2j Δnbinj

(1)

where Δninj b represents the quantity of the polymer added per injection. This quantity is measured in basemol (moles of the repeat units of the polymer). In the case of PEG, one basemol has the mass of 44 g. Use of Q2j in eq 1 is somewhat incorrect since Q2j does not take into account the effect of depletion of PEO from the solution due to adsorption. However, as seen in Table S1, the dependence of Q2j on the concentration of PEO in the solution is small, and hence the associated error can be ignored. The values of ΔQ̂ Dj are listed in Table S1. The last value of ΔQ̂ Dj is less than 0.3% of the initial value indicating that at the last injection, the silica surface is practically saturated with the polymer. The integral molar heat of adsorption per basemol after j equal injections is computed as follows

Figure 1. Typical thermogram obtained from titration of silica nanoparticles with aqueous PEG solution: PEG100K, T = 298.15 K, volume of the titrant per injection = 8 μL, initial weight fraction of silica in the aqueous suspension = 0.02 (w/w), concentration of PEG in the titrant = 10 g·L−1.

ΔQ̂ I =

Δnbinj nb

j

∑ ΔQ̂ D

i

i=1

(2)

where nb represent the total quantity of the polymer in the cell at the end of the injection. The values of ΔQ̂ I are listed in the last column of Table S1. The differential molar heat of adsorption, ΔQ̂ Dj, and integral heat of adsorption, ΔQ̂ I, as functions of the total amount of the polymer in the cell (nb) are plotted in Figure 3a,b. It is seen from the figures that the heat of adsorption progressively decreases with the increase in the amount of PEG added. At the beginning of the experiments, the available surface area of silica is much larger than the area covered by the injected chains. Thus, the PEG chains lie flat on the surface and try to establish many contacts with the surface, which leads to a greater evolution of heat. With the increase in surface coverage, the formation of loops and tails gradually increases due to insufficient availability of the surface area. This decreases the heat of adsorption per basemol. A minimum value is reached when the surface is saturated. Some titrations were performed in a reverse manner, that is, with the polymer solution in the cell and silica suspension in the syringe. In this case, 0.02 (w/w) silica from the syringe was injected into 10 g·L−1 solutions of PEG20K in the cell at 298.15 K. In Figure 4, panels a and b represent the thermograms for the reverse titration experiments at the speeds of agitation of 300 and 500 rpm, respectively.

Figure 2. Thermogram for titration of the supernatant solution with the aqueous PEG solution: PEG100K, T = 298.15 K, volume of the titrant per injection = 8 μL, the initial solution in the cell is the supernatant solution obtained by centrifugation of 0.02 (w/w) silica suspension, the concentration of PEG in the titrant = 10 g·L−1.

heat of dilution of PEG solution by water accompanying the silica suspension, and it is denoted by −Q2j, for the jth injection. Values of −Q1j and −Q2j obtained from Figure 1 and Figure 2, are listed in Table S1 (Supporting Information). Comparison of the values of −Q1j and −Q2j clearly show that the heat of dilution is very small compared to the heat of adsorption, except near the end of the titration when the surface is almost saturated with the polymer. The differential molar heat of adsorption for the jth injection, ΔQ̂ Dj, is calculated using the following equation

Figure 3. Two forms of presentation of the heat of adsorption of PEG: PEG100K, T = 298.15 K. Data correspond to Table S1: (a) differential heat of adsorption; (b) integral heat of adsorption. C


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Figure 4. Thermogram for titration of aqueous PEG solution with silica nanoparticle suspension (reverse titration) at stirring speeds of (a) 300 rpm and (b) 500 rpm: PEG20K, T = 298.15 K, volume of the titrant per injection = 10 μL, initial concentration of PEG in the cell = 10 g·L−1, weight fraction of silica in the titrant = 0.02 (w/w).

Comparison of Figure 4 with Figure 1 reveals that the heat evolved per injection in the reverse titration is much lower than the corresponding amount evolved during the straight titration. This is expected since the number of silica particles added per injection was small, and consequently, the amount of the polymer adsorbed was also small compared to that during the straight titration where a large number of particles are present in the solution. However, the important observation from Figure 4a (speed of agitation 300 rpm) is that the sizes of the peaks vary in the erratic manner. At the higher speed of agitation (Figure 4b), the peaks are more uniform. A possible reason for this behavior is that the flocculation of silica particles occurs during the reverse mode of titration. In this case, the injected particles encounter a high concentration of PEG. The adsorption is therefore very rapid and polymer chains tend to bridge the particles before they can segregate from each other by hydrodynamic forces. The flocculation creates hindrance to further adsorption and hence there is incomplete coverage of the surface area by the polymer, yielding lower heat of adsorption. At the higher speed of agitation (500 rpm), particles spread out faster and the extent of flocculation is lower. Peaks are therefore more regular. However, some flocculation does occur even at 500 rpm as is evident from the fact that the heat evolved per unit area of the particles, obtained from the reverse titration (7.75 mJ.m−2), is less than that obtained from the straight titration (8.52 mJ.m−2). It was therefore concluded that the data obtained from the reverse titration are inaccurate. Consequently, this mode of titration was abandoned. That there was no flocculation of silica particles during the straight titration was verified by measuring the particle size before and after the adsorption. The particle size was measured using dynamic light scattering (DLS: Brookhaven BI-200SM). The samples for the measurement of particle sizes were prepared by controlled addition of 10 g·L−1 PEG solution to 0.02 (w/w) silica nanoparticle suspensions under uniform stirring until particle were saturated with the polymer. The measurements were repeated three times to verify the reproducibility. Table 1 shows the effective diameter of silica particles after adsorption of PEG, as measured by DLS. It is seen from the table that the particles have remained monodisperse after adsorption, indicating that flocculation has not occurred during the experiment. The increase in the radius of the particles is about 2.5 nm, which approximately corresponds to the hydrodynamic thickness of the adsorbed layer of the polymer.28

Table 1. Effective Diameter of Silica Particles after Adsorption of PEG. Measurement Using Dynamic Light Scattering Dp before Adsorption of PEG = 20.1 ± 0.4 nm polymer

Dp (nm)

PEG8K PEG20K PEG100K

22.4 ± 0.4 23.2 ± 0.4 25.0 ± 0.4

2.2. Estimation of Adsorption Isotherms of PEG on Silica Nanoparticles. The adsorption isotherms were determined using the depletion method. A solution containing the known amount of PEG was gradually added to a silica suspension under vigorous stirring. The amount of PEG added was in excess over that required for saturation of the silica surface. At the end of the addition, the suspension was centrifuged to separate the particles. The concentration of PEG in the supernatant solution was measured to estimate the surface excess of PEG, by the difference. A calorimetric technique was used for the estimation of PEG in the supernatant solution. The details of this technique are described below. For the purpose of measurement of PEG concentration in the supernatant solution, 0.25 (w/w) silica suspension in water (in the cell) was titrated against the solution. The high concentration of suspension was chosen because it offers a large surface area for adsorption. Moreover, only a small amount of the polymer was added during the titration. This not only ensures that the entire polymer in the titrant is adsorbed, but also that there is no overlap of the adsorbed chains on the solid surface. This procedure produces titration peaks with identical area, over several injections. Also, the peak area depends only on the quantity of polymer injected, irrespective of the concentration PEG in the titrant. Calibration runs are conducted to relate the concentration of the polymer in the titrant to the heat evolved during the titration. A typical thermogram obtained during the calibration experiment is shown in Figure 5a. The volume of the titrant per injection was fixed at 4 μL in all experiments. Areas under the peaks numbered 2 to 6 are averaged to obtain the value of Q1 (the heat evolved per injection). The values of Q1 are plotted against the concentration of PEG in the titrant in Figure 5b. It is seen that the plot is a straight line passing through the origin. This indicates that the amount of the heat evolved is directly proportional to the quantity of the polymer injected. The slope of this plot was used for the estimation of unknown D


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Figure 5. (a) Typical calibration thermogram obtained from titration of silica nanoparticles with aqueous PEG solution; (b) Calibration curve for PEG: PEG20K, T = 298.15 K, volume of the titrant per injection = 4 μL, initial weight fraction of silica in the aqueous suspension = 0.25 (w/w), concentration of PEG in the titrant = 1 g L−1.

concentration. This slope was found to depend on the molecular weight of the PEG. This dependence is depicted in Table 2. Table 2. Slopes of the Calibration Curves for Different Molecular Weights of PEG polymer

slope μJ (g L−1)

PEG8K PEG20K PEG100K

290.3 279.8 266.2

The decrease in the slope with the increase in the molecular weight of the polymer is indicative of the fact that the fraction of the polymer chain segments in direct contact with the surface (i.e., the train fraction) decreases with increase in the molecular weight. The train segments are the major contributors to the heat of adsorption since they directly interact with the solid surface. Adsorption isotherms for different molecular weights of PEG were obtained at 298.15 K using 0.02 (w/w) silica nanoparticle suspension. A polymer solution of known concentration was pumped into a known volume of the suspension in a beaker at low addition rates using a peristaltic pump. The suspension was kept under stirring during the addition. The quantity of the polymer added was in large excess over that required for saturation. After the addition, the suspensions were kept under stirring for 24 h. The solution was then centrifuged in a tube centrifuge at 11000 rpm to separate the silica particles. The concentration of PEG in the supernatant solution was then measured using calorimetry. The surface excess of PEG was calculated using the following equation

Figure 6. Adsorption isotherm of PEG at 298.15 K. Experimental data are indicated as follows: (■) PEG8K, (▲) PEG20K, and (●) PEG100K, (○) data reported by Fu and Santore29 [PEG molecular weight = 120 kDa].

studied. It is also seen that our data for PEG100K are in good agreement data of Fu and Santore29 for the 120 kDa molecular weight of PEG.

3. THE MODEL The model, which relates the integral heat of adsorption ΔQ̂ I with the polymer surface interaction parameter χ*, is developed in three stages. We first derive the relationship between the integral heat of adsorption and the excess free energy of the interface γ. Next, we relate the concentration profile of the polymer in the interfacial region to the interaction parameter χ*. Next, we express γ in terms of the volume fraction profile of the polymer in the interfacial region. This establishes the connection between χ* and the integral heat of adsorption ΔQ̂ I. In the last part of this section, we describe the procedure for estimating χ* from the experimental data of ΔQ̂ I. 3.1. Relating Integral Heat of Adsorption to Excess Interfacial Free Energy. The first law of thermodynamics for a closed system can be written as δQ = dU + δW (4)

ΔM (3) A where ΔM denotes the amount of PEG adsorbed and A denotes the surface area of the silica particles used. Figure 6 shows the adsorption isotherm of three different molecular weights of PEG. The figure also compares our data with those reported by Fu and Santore.29 It is seen from the plot that adsorption isotherms are strongly dependent on the molecular weight of the polymer but weakly dependent on the concentration of the solution, in the range of the concentrations Γ=

where δQ is the heat input to the system, dU is the change in the internal energy and δW is the work done by the system. If the E


ÄÅ É ÅÅ ∂(γw /T ) ÑÑÑÑ 2 Å Å ÑÑ ΔQ 1 = AÅÅγw + T ÅÅÇ ∂T ÑÑÑÖ

Industrial & Engineering Chemistry Research work done by the system is confined only to volume change (i.e., = P dV) and if the pressure is constant and the volume change is negligible, then δQ = dU = dH. This equation is valid for the mixing process for which δQ m = dHm

where dHm is the enthalpy of mixing. The aforementioned equation is valid for the differential process. For an integral process, the following form is obtained by integration of the equation

n

ΔQ 2 = ΔHm =

(6)

(7)

∂(ΔGa /T ) ∂T

ΔQ 3 = nbsΔĤdb + n ws ΔĤdw (14) s s ̂ ̂ Here nb = rbnp. ΔHdb and ΔHdw are the specific enthalpy

(8)

For a system containing solid particles in equilibrium with an aqueous solution of polymer, the excess free energy per unit area of the interface γ is defined by the following equation γ=

1 [(n psμps + n ws μws )e − (n psμpb + n ws μwb )e ] A

(13)

where Q2j values are those obtained from Figure 2 (values are listed in Table S1). 3. Transferring nsw water molecules and nsp Kuhn segments (more precisely, polymer chains corresponding to nps Kuhn segments) from the solution obtained in step 2 to an infinitely large volume of the solution kept at the equilibrium solution concentration (i.e., corresponding to state e). The heat to be transferred to the system in order to accomplish this task is given by

If the adsorption is performed quasi-statically, then dWa = −dGa, where dGa is the Gibbs free energy of adsorption. Using the Gibbs−Helmholtz equation, we can relate the enthalpy of adsorption to dHa with the corresponding free energy in the integral form and rewrite eq 7 as ΔQ a = −ΔGa − T 2

∑ Q 2j j=1

When adsorption occurs, the work of adsorption δWa is performed by the system hence for this process δQ a = δWa + dHa

(12)

In this equation, γw represents the interfacial tension between the silica surface and pure water. 2. Mixing the injected polymer solution with the water in the cell. The heat to be added is the heat of mixing. If the given amount of polymer solution is injected in say n injections, then

(5)

ΔQ m = ΔHm

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of dilution from the initial solution (that obtained at the end of step 3) having the concentration ϕpi (volume fraction of polymer) to the solution at the equilibrium, having concentration ϕpe. ΔĤ db is defined per basemol of polymer and ΔĤ dw is defined per mole of water. 4. Transferring nsw water molecules and nsp Kuhn segments from the solution obtained at the end of step 4 to the solid−liquid interface. The heat input for this step is ΔQ4 = ΔQa, which is defined by eq 10. The total heat input for the entire process is obtained by the addition of heat inputs of the individual steps. Thus, we can write the expression for the integral heat of adsorption as ÄÅ É γ − γw Ñ ÅÅÅ ÑÑÑ ∂ ÑÑ AÅ T ÑÑ ΔQ̂ I = − ÅÅÅÅ(γ − γw ) + T 2 nb ÅÅÅ ∂T ÑÑÑÑ ÅÇ ÑÖ s s n n 1 + ΔHm + b ΔĤdb + w ΔĤdw nb nb nb (15)

(9)

where A is the total area of the solid−liquid interface. The superscripts s and b correspond to species on the solid surface (adsorbed species) and the bulk solution respectively, whereas the subscripts p and w represent polymer segment (Kuhn segment) and water molecule, respectively. The subscript e corresponds to the equilibrium state. From eq 9, it is clear that the excess free energy is the virtual work (per unit area of the interface) of transferring the nsp Kuhn segments of the polymer and nsw moles of water from the bulk solution to the solid−liquid interface at constant temperature and pressure, keeping the bulk solution in the equilibrium with the solid. We can, therefore, replace ΔGa in eq 9 by γA and rewrite as ÄÅ ÉÑ Ñ ÅÅÅ 2 ∂(γ / T ) Ñ ÑÑ ΔQ a = −AÅÅγ + T ÅÅÇ ∂T ÑÑÑÖ (10)

(

)

3.2. Continuum Model for Determining Concentration Profile of Polymer in the Adsorbed Layer. The solid−liquid interfacial region is spatially inhomogeneous, that is, its composition varies with distance from the interface. The continuum form of the self-consistent field theory was used to estimate the concentration profile of the polymer.30 The model is briefly described here. We consider a planar solid−liquid interface. The actual surface of the particle is curved. However, if the radius of the particle is much larger than the Kuhn length of the polymer, the curvature of the particle is not expected to influence the equilibrium adsorption characteristics. The Kuhn length of PEG is 0.7 nm, which is much smaller than the radius of the particle (10 nm). Hence, the curvature will not have a significant effect on the adsorbed amount as well as concentration profile. The direction along the normal to the interface is denoted by z and the surface itself forms the xy-plane. The adsorbed layer is assumed to extend from z = 0 (solid surface) to some distance z

If each Kuhn segment contains rb basemol, then np is related to nb by the following equation n np = b rb (11) We consider an experiment in which, initially, a mixture of water and silica particles is placed in the cell of the calorimeter and then the solution, containing the total of nb basemol of polymer, and nw moles of water is injected into the cell, keeping both the pressure and the temperature of the system constant. We view this process in the following steps: 1. Transferring the adsorbed water molecules from the solid surface to the bulk water in the cell. The heat to be added into the system for carrying out this step is given by F


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Industrial & Engineering Chemistry Research = δa. The concentration of the polymer in the adsorbed layer is assumed to be uniform in the x and y directions but to vary in the z direction. For the convenience of the analysis, the adsorbed layer is divided into two regions, that is, the surface phase and the interphase. The surface phase is at z = 0 and consists of the polymer segments and the solvent molecules, which are in direct contact with the solid surface. We denote the surface phase by a superscripted asterisk (∗). We assume it to be a region having zero thickness but holding a finite mass. The concentrations in the surface phase are measured in terms of area fractions: φp* for the polymer and φw* for water (φp*+ φw* = 1). The interphase extends from z = 0 to z = δa. We measure the concentrations of species in the interphase in terms of volume fractions: ϕp(z) for the polymer and ϕw(z) for water (ϕp(z) + ϕw(z) = 1). The surface phase meets the interphase at z = 0, and the two are assumed to be in local equilibrium at that location. The direct interaction between the solid surface and the solution is confined only to the surface phase. The configurational statistics of a sample chain in this field is obtained by solving the connectivity equation for the chain30 S

∂Gp(z , q) ∂q

The volume fraction of the polymer at location z in the interphase is related to bulk volume fraction by the following equation: ϕp(z) =

φp* =

where Gp(z,q) is the probability of locating the end segment of a polymer subchain with contour length q(0 < q < rl), at location z, relative to the bulk. Equation 16 is solved subject to the following boundary conditions

Gp(δa , q) = 1

(18)

l

∂Gp(0, q) ∂q

ÅÄÅ Å l ∂Gp(z , q) ) ÅÅ ∂z ÅÅÅ 2 ÅÇ ÉÑ ÑÑ Ñ * + 2e(up(0) − up )/(RT )}ÑÑÑÑ ÑÑ ÑÑÖ

ÅÅ (u p(0) − u p*)/(RT ) −1 Å

= (1 + 2 e

+ Gp(0, q){1 − 2 eup(0)/(RT )

z=0

Gp(z , q)Gp(z , rl − q) dq

(21)

ϕpb rpl

*

e up / RT

∫0

rl

Gp*(q)Gp*(rl − q) dq

(22)

∫

Equation 17 states that the probability weight of the end of a subchain of zero contour length is entirely governed by its potential since this end is free from any polymer attachment. Equation 18 is a reminder of the fact that probability Gp is weighted with respect to the bulk solution (z = δ), hence its value in the outer boundary of the adsorbed layer is unity. Equation 19 is the boundary condition at z = 0 and is derived by taking into account the spatial constraint imposed by the solid surface.30 A segment at z = 0 can either be detached from the surface or attached to the surface. The corresponding energies up(0), up* are different since the latter include the energetic interactions with the molecules of the surface. The probability G*p (q) of finding the end of a subchain of length q in the surface phase can be related to its probability at z = 0 (in the unattached state) as

Gp(0, q)

rl

b b jij ϕp μp ϕ bμ b zyz F = V bjjjj + w w zzzz + V bζ b(ϕpb + ϕwb − 1) jj vp vw zz k { Ä Å z ϕ ( ) δa Å ϕ (z)μw (z) ÅÅÅ p ÅÅ +A (μph (z) − εp(z)) + w 0 Å vw ÅÅÅÇ vp ÑÉ ÅÄÅ * ÑÉÑ ÅÅ φp ÑÑ φ*μ* ÑÑÑ + ζ(ϕp(z) + ϕw (z) − 1)ÑÑÑÑ dz + AÅÅÅÅ (μp*h − εp*) + w w ÑÑÑÑ ÑÑ a w ÑÑÑ ÅÅÅ a p ÑÖ ÑÖ ÅÇ * * + Aζ *(φ + φ − 1)

(19)

Gp*(q)

∫0

3.3. Relation between the Interfacial Free Energy and the Composition of the Adsorbed Layer. The excess surface free energy of polymer γ and water γw are related to the composition of the adsorbed polymer layer through chemical potential expressions. The model used to derive this relation is based on the treatment proposed by Evans and Needham31 and is described in detail in our previous work.28 We only provide a brief account here. The chemical potential of a polymer segment in the adsorbed layer consists of two contributions. The local part, μhp(z), arises due to local environment at z and can be calculated using Flory− Huggins theory, assuming the solution to be locally homogeneous. The nonlocal contribution −εp(z) arises by the virtue of the fact that the polymer segment at z is a part of the chain which is adsorbed on the solid surface.28,30 Thus, the attractive pull of polymer−surface interaction is imparted to the segment at the location z through the subchain connecting the segment to the solid surface. The water molecules, as well as the polymer chains in the bulk, have no nonlocal contribution associated with them. The expression for the total Helmholtz free energy functional of the system includes the contributions from the bulk phase, the interphase, and the surface phase. The constraints are introduced into the functional using Lagrange multipliers.

(16)

(17)

rpl

eup(z)/ RT

The corresponding expression for the area fraction of the polymer in the surface phase is

2 S2 ∂ Gp(z , q) = + [1 − eup(z)/(RT )]Gp(z , q) 6 ∂z 2

Gp(z , 0) = e−up /(RT )

ϕpb

p

w

(23)

The first two terms correspond to the bulk solution, to which we assign superscript b. The third (integral) term corresponds to the interphase (without superscript) and the last two terms correspond to the surface phase (superscript *). Vb is the volume of the bulk solution and A is the area of the interface, δa is the thickness of the adsorbed layer (which is arbitrarily fixed by locating its outer boundary at a point at which polymer concentration is nearly equal to the bulk concentration). The terms vp and vw are the partial molar volumes of the polymer segment and the water molecule, respectively, and ap and aw are the partial molar areas occupied by the respective species in the surface phase. The contribution to local potential in the interphase and surface phases are denoted by μhp(z) and μp*h, respectively, and the corresponding nonlocal contributions are denoted by − εp(z) and − ε*p .

*

= e−[up − up(0)] /(RT ) (20) G



ÅÄÅ δϕ (z) ÅÅ h ÅÅ(μ (z) − μ b − ε (z)) p ÅÅ p p p 0 Å vp ÅÅÇ ÉÑ ÑÑ δϕ (z) Ñ + (μw (z) − μwb ) w + ζ(δϕp(z) + δϕw (z))ÑÑÑÑ dz ÑÑ vw ÑÖ ÄÅ ÉÑ ÅÅ * ÑÑÑ δφp* δφ ÅÅ h + AÅÅÅ(μp* − μpb − εp*) + (μw* − μwb ) w ÑÑÑÑ ÅÅ ap a w ÑÑÑ ÅÅÇ ÑÖ + Aζ *(δφp* + δφw*) = 0

Apart from the contributions to the Helmholtz free energy, we must also include the space filling constraints (the sum of the volume fractions of polymer and water is unity in the bulk phase as well as in the interphase and the sum of the area fractions of the species is unity in the surface phase), which are accompanied by the Lagrange multipliers ζb, ζ and ζ*. The Lagrange multiplier for the confinement constraint is denoted by τ. The unknowns in eq 23 are obtained by variational minimization of the free energy functional; that is, we write δF = 0. This gives

ζ b(δϕpb + δϕwb) + A

ζb = 0

∫

1 * (μ − μwb ) + ζ * = 0 aw w

(i) Gibbs−Duhem equations in the bulk, the interphase, and the surface phase, respectively. At constant temperature and pressure, these equations can be written in terms of variations of the volume fractions

ϕp(z) vp φp* ap

vw

(25)

(δμph (z) − δεp(z)) +

(δμp*h − δεp*) +

φw* aw

ϕw (z) vw

δμw (z) = 0

εp* = μp*h − μpb −

(ii) Conservation of the total volume (volume of the interphase plus that of the bulk solution), (28)

δV

ϕkb

vk

+A

∫0

δa

δϕk (z) vk

dz + A

ϕkb vk

δ(δa) + A

δφk* ak

(μw (z) − μwb )

(μw* − μwb )

(34)

(35)

If all the molecules in the adsorbed layer are placed in the bulk solution, the resulting Helmholtz free energy (Fb) can be derived from eq 36 by replacing each of μw(z) and μw* by μbw

(iii) Conservation of the total number of moles of each species in the system b

aw

vw

∫

(27)

Aδ(δa) + δV b = 0

ap

vp

The expression for the total Helmholtz free energy can now be obtained from eq 23 by eliminating the nonlocal potentials using eq 34 and dropping the terms containing the Lagrange multipliers. ÄÅ i ϕb ϕpb zyz δa Å j v y ÅÅ ϕp(z) ji b j ÅÅ jjμ − p μ b zzz F = V bjjjj w μwb + μpb zzzz + A ÅÅ j p j j vw 0 Å vw w zz{ vp z ÅÅÇ vp k k { ÄÅ ÉÑ ÉÑ ÅÅ φ* i a p b yz μw (z) ÑÑÑÑ μw* ÑÑÑÑ ÅÅ p jj b ÑÑ ÑÑ dz + AÅÅÅ jjμp − + μ zz + ÅÅ a p jk vw ÑÑÑ a w w zz{ a w ÑÑÑ (36) ÑÖ ÅÇ ÑÖ

(26)

δμw* = 0

(33)

By eliminating ζ and ζ* from the set of eqs 32 and 33, respectively, we obtain the following expressions for the nonlocal potentials εp(z) = μph (z) − μpb −

δμwb = 0

(32)

1 *h (μ − μpb − εp*) + ζ * = 0, ap p

The following additional equations must also be considered.

vp

(31)

1 (μ (z) − μwb ) + ζ = 0 vw w

(24)

δμpb +

(30)

1 h (μp (z) − μpb − εp(z)) + ζ = 0, vp

∫

ϕwb

δa

Since each of the variations is now independent of the others, eq 30 implies that the coefficient of each is zero. This yields the following equations

ij δ(V bϕ bμ b ) δ(V bϕwbμwb ) yzzz jj p p jj zz + ζ b(δϕpb + δϕwb) + jjj zzz vp vw k { ÄÅ ÉÑ ÅÅ δ[ϕ (z)(μ h (z) − ε (z))] Ñ δa Å δ[ϕw (z)μw (z)] ÑÑÑ p p ÅÅ p ÑÑ dz ÅÅ +A + ÑÑ 0 Å vp vw ÅÅ ÑÑ ÅÇ ÑÖ ÄÅ b b ÉÑ ÅÅ ϕ μ b bÑ Ñ δ ϕ μ ÑÑ a ÅÅ p p + Aδ(δa)ÅÅÅ + w w ÑÑÑ + Aζ δϕp(z) + δϕw (z) dz ÅÅ v ÑÑ 0 v w Ñ ÅÅÇ p ÑÖ ÅÄÅ ÑÉÑ h δ(φw*μw*) ÑÑÑ ÅÅÅ δ[φp*(μp* − εp*)] Å ÑÑ + ζ *(δφ* + δφ*) = 0 + AÅÅ + Ñ p w ÅÅ ap a w ÑÑÑ ÅÅÇ ÑÖ

ϕpb

∫

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ÄÅ ÉÑ ÅÅ ϕ (z) i vp b yz μwb ÑÑÑÑ ÅÅ p jj b z ÑÑ dz jμ − μw zz + F = V jj + +A ÅÅ jj vw ÅÅ vp jjk p 0 Å vp z vw z{ vw ÑÑÑ ÅÇ ÑÖ k { É ÅÄÅ * Ñ bÑ ÅÅ φp ji Ñ a μ Ñ p by z + AÅÅÅÅ jjjμpb − μw zzz + w ÑÑÑÑ z a ÑÑ ÅÅ a p j a wÑ w ÅÅÇ k { (37) ÑÖ b

=0 (29)

For each of k = p and k = w. Using the relations shown in 25 we can simplify eq 24, to the following form

ij ϕ b bj jj w

μwb

ϕpb

yz

z μpb zzzz z

∫

δa

Hence the surface excess free energy per unit area of the interface, γ, is H


Article

Industrial & Engineering Chemistry Research F − Fb γ= = A

∫0

δa

by ij μ (z) − μ b yz i * jj w z dz + jjj μw − μw zzz wz jj zz jj z j z j a w zz vw k { k {

where χb is the χ parameter in the bulk solution. In the analogous manner, we can write the chemical potential of polymer and water in the surface phase in terms of the area fractions. Thus (38)

μp* − μp0

Since water is monomeric, it adsorbs as a monolayer in the pure state (i.e., remains only in the surface phase) and hence we can write γw as γw =

RT

μw0 * − μw0 aw

(39)

μw* − μw0

where the superscript 0 corresponds to pure water and the asterisk (∗) corresponds to the surface phase. Subtracting eq 38 from eq 39, we obtain γ − γw =

0* y ij μ (z) − μ b yz i * jj w z dz + jjj μw − μw zzz wz jj zz jj zz j z j aw z vw k { k { ij μ b − μ0 yz wz zz − jjjj w j a w zz k {

∫0

RT

δa

RT

μw − μw0 RT

=

ij 1 vp yz vp 1 ln(ϕp) + jjjj − zzzzϕw + χϕw2 j rp rp vw z vw k { vp ∂χ + ϕw2ϕp vw ∂ϕp

ij v yz ∂χ = ln(ϕw ) + jjjj1 − w zzzzϕp + χϕp2 − ϕw ϕp2 j rpνp z ∂ϕp k {

(40)

μw0 * − μw0 RT

RT

μwb − μw0 aw

= χw*

(47)

ÄÅ ÅÅ ÅÅ i y i y ÅÅ jj ϕw zz jj γ − γw 1 ÅÅlnjj zz + jj1 − vw zzzz(ϕ − ϕ b) + χ (ϕ2 − (ϕ b)2 ) = Å jj b zz jj p p p Å RT vw 0 ÅÅÅ ϕw rpνp zz p { ÅÅ k { k ÅÇ É ij yzÑÑÑÑ jj z i y jj ∂χ zz zzzÑÑÑÑ j ∂χ zz zzÑÑ dz − jjjjϕw ϕp2 − ϕwb(ϕpb)2 jjjj z jj j ∂ϕp zz bzzzzÑÑÑÑ ∂ϕp jj k {φp zÑÑ k {ÑÖ ÅÄÅ ÅÅ Å a w zyz 1 ÅÅÅÅ jijj φw* zyzz jijj b + zz(φ* − ϕw ) ÅÅlnjjj b zzz + jjj1 − Å a w ÅÅ ϕw rpa p zz p { ÅÅ k { k ÅÇ É ij yzÑÑÑÑ jj z i y z jj ∂χ zz zzÑÑÑÑ j ∂χ ′ zz zzÑÑ + χ ′ (φp*2 − (ϕpb)2 ) − jjjjφw*φp*2 − ϕwb(ϕpb)2 jjjj z j ∂ϕp zz bzzzzÑÑÑÑ ∂φp* jjj k {φp zÑÑ j k {ÑÖ

∫

(41)

ij 1 vp yz vp 1 ln(ϕpb) + jjjj − zzzzϕwb + χ b (ϕwb)2 j z rp r vw vw k p { ij ∂χ yz vp j zz + (ϕwb)2 ϕpbjjj z jj ∂ϕ zzz vw k p {φpb (43)

ji v zy = ln(ϕwb) + jjjj1 − w zzzzϕpb + χ b (ϕpb)2 j rpvp z k { ij ∂χ zy j zz − ϕwb(ϕpb)2 jjj zz jj ∂ϕp zz b k {ϕp

(46)

Substitution of the relevant expressions for the chemical potentials in eq 40 yields the following equation for γ−γw in terms of the volume fraction of the polymer

where χ is the (concentration dependent) Flory−Huggins parameter of the polymer−solvent interaction and rp is the number of Kuhn segments per chain. In the bulk solution, eq 41 and 42 reduce to =

ji a zy = ln(φw*) + jjjj1 − w zzzzφp* + χ ′(φp*)2 j rpa p z k { ′ ∂ χ + χw* − φw*(φp*)2 ∂φ*

In the above equation, χ′ represents the Flory−Huggins parameter in the surface phase and χ*p , χ*w are the free energy of interaction between polymer−solid and water−solid (normalized with RT). In general, χ′ may differ from χ due to the presence of the solid. For pure water (φ*p → 0), eq 46 reduces to

(42)

μpb − μp0

ij 1 a p yzz a 1 zzφ* + p χ ′(φ*)2 ln(φp*) + jjjj − p w z j z rp r aw aw k p { ap ap ∂χ ′ (φw*)2 φp* + + χ* aw aw p ∂φp* (45)

p

The term γ−γw is needed for estimating the integral heat of adsorption. 3.4. Relation between Interfacial Free Energy and Polymer−surface Interaction Parameter χ*. The chemical potentials are related to the volume fraction of species using the modified form of the Flory−Huggins model, which accounts for the concentration dependence of χ. Thus, in the interfacial region, we have μp − μp0

=

δa

(48)

The mean potential of the polymer segment is obtained by subtracting the entropic contribution from the nonlocal part of the chemical potential.30 Thus i y RT jjj ϕp(z) zzz lnjjj b zzz = −(μph (z) − μpb ) rp j ϕp z k { i y vp RT jjj ϕp(z) zzz − (μw (z) − μwb ) − lnjjj b zzz vw rp j ϕp z k {

u p(z) = εp(z) −

(44)

(49)

and I


Article

Industrial & Engineering Chemistry Research u p* = εp* −

i *y RT jjj φp zzz lnjjj b zzz rp j ϕp z k {

= (μp* − μpb ) −

ap aw

(μw* − μwb ) −

i *y RT jjj φp zzz lnjjj b zzz rp j ϕp z k {

to compute the integral heat of adsorption ΔQ̂ I to the first approximation. We compare the computed value with the measured value ΔQ̂ I. We update the value of χ*, until the difference between the two values is at a minimum. To conduct the above calculations, it is necessary to know the concentration of the polymer in the bulk solution, in equilibrium with the solid. Out of the total amount of the polymer injected into the cell, a part is adsorbed and the rest remains in the bulk solution. The material balance of the polymer dictates that

(50)

Substituting the expressions for the chemical potentials from 41 into eq 49, we get ÄÅ u p(z) vp ÅÅÅÅ ∂χ = ÅÅÅχ (ϕw − ϕp) − χ b (ϕwb − ϕpb) + ϕw ϕp Å RT vw ÅÅ ∂φp ÅÇ ÉÑ ij ϕ yzÑÑÑ ∂χ b − ϕwbϕpb b − lnjjjj wb zzzzÑÑÑÑ j ϕ zÑÑ ∂ϕp k w {ÑÑÖ (51) In a similar manner, we can convert following form ÅÄ u p* a p ÅÅÅÅ ÅÅχ ′(φ* − φ*) − χ b (ϕ b = w p w RT a w ÅÅÅÅ ÅÇ ÑÉÑ b jij φw* zyzÑÑÑ b b ∂χ − ϕw ϕp b − lnjjj b zzzÑÑÑ − j ϕ zÑÑ ∂φp k w {ÑÑÖ

Γp =

ϕpb

∂χ ′ − ϕpb) + φw*φp* ∂φ* χ* (52)

(58)

(59)

ϕbp,

Thus, the value of which is chosen for the estimation of χ* should satisfy this equation. There are situations, where, ϕbp is extremely small and cannot be estimated correctly by eq 59. In such cases, we adopt an alternative technique to determine χ*. We note that in these cases, practically all the added polymer is on the solid surface. In this case, eq 57 simplifies to

(53)

We assume no density change accompanying the exchange of molecules between the solution and the surface phase. This requires that ap vp = aw vw (54)

Γp = nb /Arb

The excess amount of the polymer in the adsorbed layer (in the mole of Kuhn segment) per unit area of the surface is given by ÄÅ ÉÑ ÅÅ δ i ϕ (z) − ϕ b y φp* ÑÑÑÑ zz ÅÅ a jj p p jj zz dz + ÑÑ Γp = ÅÅÅ zz Ñ ÅÅ 0 jjj z vp a p ÑÑÑ ÅÅÅ ÑÖÑ (55) k { Ç

(60)

Thus, Γp is known a priori. Moreover, we can modify the Helmholtz free energy functional by adding the confinement constrained in eq 23.28

ÄÅ ÅÅ ϕ (z) ϕ (z)μw (z) ÅÅ p ÅÅ (μph (z) − εp(z)) + w ÅÅ vp 0 Å vw ÅÇ Ä ÉÑ ÑÉ Å ÅÅ φ* ÑÑ φw*μw* ÑÑÑÑ ÅÅ p ÑÑ h ÑÑ Å * * Ñ + ζ(ϕp(z) + ϕw (z) − 1)ÑÑ dz + AÅÅ (μp − εp ) + ÅÅ a p ÑÑ a w ÑÑÑÑ ÅÅÇ ÑÖ ÑÖ * ij y ϕp zz δ ϕp j zz + Aζ *(φp* + φw* − 1) + τAjjjjΓp − dz + z j 0 vp a p zz (61) k {

∫

Vb 0 μ +A F= νw w

The first and second term of eq 55 corresponds to the amount of polymer in the interphase and surface phase, respectively. On the mass basis, the adsorbed amount (mg·m−2) is obtained by multiplying Γp by the molecular mass of Kuhn segment of PEG, which is 70.86 × 103 mg. Γ = 70.86 × 103Γp

rbV l

∫

where, χ* is the polymer−surface affinity parameter and is defined as χ * = χw* − χp*

=

nblvp

where, Vl is the volume of the liquid in the cell. Combining eq 55, 57, and 58, we obtain the equation expressing the material balance in terms of the polymer concentration profile. ÅÄÅ ÑÉ ÅÅ δa ji ϕ (z) − ϕ b zy (nb /rb) − (ϕpbV l /vp) φp* ÑÑÑÑ jj p p z ÅÅ ÑÑ j zzz dz + = ÅÅ Ñ zz ÅÅ 0 jjj A vp a p ÑÑÑ ÅÅÅ ÑÑÖ k { Ç

p

aw

(57)

where, nb represents the total number of basemols of the polymer in the cell, and nlb is the number of basemols of polymer in the bulk solution at equilibrium. The latter is related to the bulk concentration ϕbp by the following equation.

eq 50, using 43, to the

ap

nb − nbl Arb

∫

δa

∫

(56)

3.5. Computation Procedure for Estimation of χ*. The set of eq 16−20 is solved in conjunction with eq 21 and 22, to obtain the concentration profile of the polymer in the adsorbed layer. This requires the prior knowledge of the interaction parameter χ* which appears in the expression for up* (eq 52). The method we follow is that we guess the value of χ* and then predict the concentration profile of the polymer in the adsorbed layer using the set of eq 16−22. We use this concentration profile in eq 48 and estimate (γ − γw)/RT. The heat of dilution of PEG solution in the absence of solid is calculated using the methodology described by Mohite and Juvekar.32 This allows us

After the variational minimization of the free energy functional, the following expressions for the nonlocal contributions to the chemical potential are obtained. vp εp(z) = μph (z) − μw (z) − τ vw (62) εp* = μp*h −

ap aw

μw* − τ

(63)

The free energy of the adsorbed layer is J


Article

Industrial & Engineering Chemistry Research F=A

∫0

δa

μw (z) vw

ij μ* yz dz + Ajjjj w zzzz + Aτ Γp j aw z k {

4. RESULTS AND DISCUSSION 4.1. Regression of Calorimetry Data for Estimation of χ*. Except for interaction parameters χ*, χ, and χ′, all the other parameters of the model have been obtained from the literature and are listed in Table 3. PEG-water interaction parameters in

(64)

For calculation of the mean potential, we choose an arbitrary reference state with a very low concentration of polymer, which we call the bulk phase. Equation 51 and 52 are then modified to ÄÅ ÉÑ ÑÑ u p(z) vp ÅÅÅÅ Ñ ∂χ τ b Å = ÅÅχ (ϕw − ϕp) − χ + ϕw ϕp − ln ϕw ÑÑÑÑ − ÑÑ RT RT vw ÅÅÅ ∂ϕp ÑÖ Ç

Table 3. Parameters Used in Mean Field Calculation. System: PEG−Water−Silicaa parameter

(65)

ÄÅ ÉÑ ÑÑ a p ÅÅÅÅ ′ Ñ ∂ χ *ÑÑÑ ÅÅχ ′(φw* − φp*) − χ b + φw*φp* ln − φ = wÑ Å * Å ÑÑ RT a w ÅÅ ∂φp ÑÖ Ç ap τ − χ* − aw RT (66) u p*

i3y interfacial area occupied by water in a w = jjj zzz the surface phase, aw28 k4{

2/3

π 1/3vw 2/3Nav1/3m 2·mol−1

a

Note: Molecular weight of Kuhn segment = 70.86, no. of repeat units per Kuhn segment, rb = 1.61.

and eq 48 gets modified to

ÅÄÅ ÑÉÑ ij Ñ vw yzz 1 δa ÅÅÅÅ j 2 2 ∂χ Ñ ÑÑdz j z = zzϕp + χϕp − ϕw ϕp ÅÅln(ϕw ) + jj1 − Ñ j RT vw 0 ÅÅÅ rpνp z ∂ϕp ÑÑÑÑ k { ÅÇ ÑÖ ÄÅ ÑÉÑ ÅÅ i y Ñ jj a w zz 1 ÅÅÅ ÑÑ 2 2 ∂χ ′ Ñ j z * * * * * + Åln(φw ) + jj1 − Ñ zφ + χ ′ (φp ) − φw (φp ) j a w ÅÅÅÅ rpa p zz p ∂φp* ÑÑÑÑ k { ÅÇ ÑÖ τ Γp + (67) RT

γ − γw

equation

number of Kuhn segments per chain, 0.0141Mp (Mp = molecular weight of PEG) rp.28 length of the Kuhn segment, l28 0.707 nm partial molar volume of water, vw28 6.377 × 10−8T + 16.15 × 10−6 m3·mol−1 ratio of the partial molar volume of 3.27 PEG segment to that of water, vp/vw28

∫

the Flory−Huggins theory are used, polynomial based on polymer volume fraction in solution phase, and polymer area fraction in surface phase given below (T is temperature expressed in Kelvin).32

It is seen from eq 67 that τ is the additional free energy possessed by the adsorbed layer, per mol of polymer segment, due to the confinement constraint. The procedure to obtain the excess free energy involves choosing an arbitrary value of the reference concentration ϕbp (we have used ϕbp = 1 × 10−16 in all cases) and guessing χ*. The value of τ is so chosen the surface excess is Γp, which is given by eq 60. The value of the excess free energy is then estimated using eq 67. The advantage of this procedure is that it eliminates the inaccuracies associated with the estimation of ϕbp by material balance. The estimate of the excess free energy is insensitive to the choice of ϕbp. Depending upon the choice of ϕbp, the value of τ adjusts in such a way that (γ − γw)/RT remains unchanged. The use of this procedure is illustrated later. The self-consistent field model presented here is based on the theory developed by Scheutjens and Fleer (SF).2,33,34 However, there are important differences between the present model and the SF model. The SF model is lattice based, where it is not possible to directly account for stiffness of the polymer chain since a single set of lattice parameters is used to characterize both the solvent and the polymer segment. Also, the lattice model cannot accurately predict sharp variations in the volume fraction profile close to the interface due to the discrete nature of the lattice. In the continuum model, both these effects can be taken into account in a straightforward manner. Also, the technique used for estimating the nonlocal chemical potential, allows us to incorporate the effect of the external constraints in a straightforward manner. Moreover, the nonlocal effect of the concentration dependence of the Flory−Huggins parameter can be accounted through energy minimization as described above. This allows us to extend the interaction models derived from the homogeneous solution to the adsorbed layer.

χ = b0(T ) + b1(T )ϕp + b2(T )ϕp2

(68)

χ ′ = b0(T ) + b1(T )φp* + b2(T )(φp*)2

(69)

where b0(T ) = 26.310 − 1.7168 × 103/T − 3.5519 ln(T ) (70) 3

b1(T ) = 24.657 − 1.5470 × 10 /T − 3.4314 ln(T )

(71)

b2(T ) = 9.0804 − 6.9445 × 102 /T − 1.2222 ln(T ) (72)

The coefficients of this polynomial bi(T) are temperature dependent. The temperature independent part of bi(T) was obtained using the activity data of PEG in water at 303.15 K, and the temperature dependent part was obtained from the data on enthalpy of mixing of PEG8K and water in the temperature range from 288.15 to 348.15 K. This correlation was tested for activity coefficients of PEG in solution and enthalpy of mixing PEG−water for different molecular weights and temperatures. It was also used to predict the phase diagram in the LCST region for different molecular weights of PEG. The match in all cases was good. This proves that bi(T) obtained by this procedure is independent of the molecular weight of PEG. The adsorption of PEG on silica, during the calorimetric titration, has two distinct regimes. During the initial phase of titration, the number of chains of PEG added into the cell is much smaller than that needed for covering the surface area of the silica particles. Hence the added chains cover only a few particles, and the rest of the particles have no PEG on them. Since the present model assumes uniform coverage of the surface by the polymer, it is not applicable in this situation. When a sufficiently large amount of the polymer is added, the surface coverage becomes more uniform and the present model is applicable. We illustrate this point below. K


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Figure 7 also compares the consistency of the two alternative calculation procedures outlined in the previous section. The first procedure uses material balance and is applicable when the significant quantity of polymer is left in the solution at the equilibrium. The solid part of the curve in the figure is based on this procedure. The other procedure, which confines the entire polymer on the solid surface, is convenient when the quantity of the polymer in solution is too small to be estimated accurately by material balance. The dotted line in Figure 7 is based on this technique. The solid and the dotted lines overlap each other without any discontinuity indicating that the two procedures are consistent with each other. There is an overlap region, where both procedures are equally accurate. This region is covered by stars in the inset of the figure. The stars themselves correspond to the second procedure (based on τ) and the underlying solid line is based on the procedure which uses the material balance. We see an excellent match between the two procedures in the overlap region, indicating that both procedures are consistent with each other. Figure 8 presents the experimental data for the integral enthalpy of adsorption of PEG. Each figure contains data for one molecular weight of PEG, but at three different temperatures. The data for low nb are omitted for the reason explained with reference to Figure 7. The points are the experimental results and the lines are the best fits of the model. The only unknown parameter in the model is χ*, which is obtained from these data by regression. The enthalpy data are very sensitive to χ*. This is demonstrated in Figure 9 in which the best fit plot is shown by the solid line and corresponds to χ* = 0.512. The dashed line corresponds to χ* = 0.522 (+2% deviation) and the dotted line corresponds to χ* = 0.502 (−2% deviation). The percent deviations of these lines from the best fit line are +8.8(χ* = 0.522) and −7.8 (χ* = 0.502). This yields the sensitivity of the heat of adsorption to χ*(d ln ΔQ̂ I/dχ*) as approximately 8. Table 4 lists the values of χ* obtained from the regression analysis of the experimental data from Figure 8a−c. The table also lists the standard deviation of the data from the regression lines. It is seen from the table that standard deviations are within 5% of the value of χ* in most of the cases, indicating that accuracy of estimation of χ* by the present model is within 1.3%. We see from the table that the value of χ* is approximately 0.5, which means that the difference in the energies of interaction

Figure 7 compares the experimental data of integral enthalpy of adsorption (hollow squares) with the prediction based on the

Figure 7. Comparison of model prediction with experiments. PEG20K, T = 298.15 K. Experimental data are indicated by the symbol □. The model prediction is done using χ* = 0.512 and is represented by the solid part of the line as well as the dotted part of the line. The solid line is based on the calculation, which incorporates material balance, and the dotted line is based on the use of the Lagrange parameter τ. The inset compares the two calculation procedures. In the overlap region, both calculation procedures are used. The stars correspond to the procedure based on τ, whereas the underlying solid line is based on material balance.

model (solid and the dotted lines), for the case of PEG20K at 298.15 K. The model uses the value of χ* = 0.512. The value is so chosen that the model fits the experimental data over a substantially large range of nb. The figure clearly shows that the model fits the experimental data well for higher values of nb, but overestimates ΔQ̂ I for lower values of nb. The reason for this discrepancy is that at very low polymer concentrations, the polymer adsorbs on the solid surface in the form of isolated single chains, which form discrete patches on the surface. Since the present model assumes that the entire surface of the solid is covered uniformly with the polymer, it estimates the enthalpy based on an assumption of a thinner adsorbed layer with a larger number of polymer segments in contact with the solid than the actual. This leads to an overprediction of the heat of adsorption. We have excluded this region from our analysis.

Figure 8. Integral heat of adsorption. Experimental data are indicated by symbols: (■) T = 298.15 K, (▲) T = 308.15 K, and (●) T = 318.15 K. The predictions are represented by lines. (a) PEG8K: () T = 298.15 K (χ* = 0.520), (---) T = 308.15 K (χ* = 0.506), (···) T = 318.15 K (χ* = 0.502). (b) PEG20K: () T = 298.15 K (χ* = 0.512), (---) T = 308.15 K (χ* = 0.497), (···) T = 318.15 K (χ* = 0.494). (c) PEG100K: () T = 298.15 K (χ* = 0.505), (---) T = 308.15 K (χ* = 0.495), (···) T = 318.15 K (χ* = 0.488). L


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We expect the value of χ* to be independent of the molecular weight of the polymer. From Table 4, we find that it increases slightly with the decrease in the molecular weight of the polymer. A possible reason is the effect of the end hydroxyl groups of the PEG chain, which forms stronger bonds with silica. The effect is pronounced for shorter chains because of the predominance of OH in shorter chains. This is consistent with the values in Table 4, which show a larger rate of increase in χ* with the molecular weight from PEG20K to PEG8K compared to that from PEG100K to PEG20K. We also see from the table that χ* decreases weakly with temperature, indicating that the enthalpic contribution to χ* is small compared to the entropic contribution. 4.2. Comparison with Calorimetry Data of Trens and Denoyel. Trens and Denoyel23 have reported calorimetric measurements of the heat of adsorption of PEG on silica particles. Figure 10a compares one of their results with the present work. The results are reported in the form of ΔQΓI , the integral heat of adsorption per basemol of the surface excess, and are shown by open squares. Our data of ΔQ̂ I are converted to Γ ΔQI b using the following equation.

Figure 9. The sensitivity of ΔQ̂ I to χ*. PEG20K at 298.15 K, Experimental data are indicated by black filled squares and model prediction by lines: () χ* = 0.512, (···) χ* = 0.502, and (---) χ* = 0.522.

Table 4. Regression Estimates of Polymer−surface Interaction Parameter, ÄÅ ÉÑ1/2 exp model Å ndata Ñ χ*σ = ÅÅÅ∑ j = 1 (ΔQ̂ Ij − ΔQ̂ Ij )2 /(n data − 1)ÑÑÑ ÅÇ ÑÖ

ΔQ IΓb =

T = 298.15 K

T = 308.15 K

T = 318.15 K

polymer

χ*

σ

χ*

σ

χ*

σ

PEG8K PEG20K PEG100 K

0.520 0.512 0.505

0.016 0.012 0.017

0.506 0.497 0.495

0.022 0.043 0.026

0.502 0.494 0.488

0.021 0.054 0.091

ΔQ̂ Inb rb ΓbA

(73)

where A represents the total surface area of the porous silica particles. These are plotted as the filled triangle in the same figure. The solid line represents the prediction from our model. It is seen from the figure that the values reported by Trens and Denoyel23 are higher than ours at low Γ, but lower at high Γ. There are two important differences in the experiments conducted by Trens and Denoyel23 from those conducted by us. First, the experimental technique used by Trens and Denoyel23 (flow calorimetry) ensures uniform coverage of PEG on the silica surface even at low values of Γ and hence realizes higher heats of adsorption than those realized in our experiments, at low Γ. Second, Trens and Denoyel23 have used porous silica particles in their study. We suspect that some of the pores in their particles are not accessible to PEG and hence a certain fraction of the surface area is not covered by the polymer. Since Trens and Denoyel23 base their calculations on the assumption that the entire surface area is covered, they predict a lower value of the heat of adsorption. To correct their result, we

between PEG (monomer)−silica and water−silica is approximately 1/2 kT. The value of χ* depends on the extent of ionization of silica. An increase in the fraction of silanol groups reduces the value of χ*.28 The estimated value of χ*, based on the procedure described in this reference is between 0.5 and 0.6, which is in agreement with the value obtained from microcalorimetry. The value compares well with the value reported by van der Beek and Cohen Stuart.35 The reported value of segmental adsorption energy of PEO on the silica surface measured by microcalorimetry is 1.2 kT at pH 5.5 by Trens and Denoyel,23 which is higher than the present value of χ*. The reason for this difference is discussed in section 4.2.

Figure 10. Comparison with the integral heat of adsorption data of Trens and Denoyel: PEG100K at 298.15 K. The filled triangles represent our data and the open squares represent the data of Trens and Denoyel.23 The solid line represents the model prediction using χ* = 0.501. (a) Uncorrected data of Trens and Denoyel; (b) corrected data of Trens and Denoyel23 after accounting for the effect of porosity. M


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Figure 11. Adsorption isotherm of PEG at 298.15 K. (a) Measured using calorimetry. Present experimental data obtained from calorimetry are indicated by symbols: (■) PEG8K, (▲) PEG20K, and (●) PEG100K. The predictions are represented by lines: () Mp = 8 kDa (χ* = 0.520), (---) Mp = 20 kDa (χ* = 0.512), (···) Mp = 100 kDa (χ* = 0.505) and (−·−·−) Mp = 120 kDa (χ* = 0.505). (b) Optical reflectivity measurements. Experimental data reported by Fu and Santore29 are indicated by symbols (□) Mp = 32 kDa, (△) Mp = 120 kDa. The predictions are represented by lines: () Mp = 32 kDa (χ* = 0.510), (---) Mp = 120 kDa (χ* = 0.505). (c) Variation with molecular weight of PEG; experimental data at 298.15 K. (■) Present data [ϕbp = 1 × 10−3], (◇) van der Beek et al.19 [ϕbp = 6 × 10−3], (△) Fu and Santore29 [ϕbp = 0.5 × 10−3], (□) Vangeyte et al.36 [ϕbp = 3 × 10−3], and (○) Mubarekyan and Santore37 [ϕbp = 5 × 10−3]. The solid line represents the prediction based on χ* = 0.512.

introduce a correction factor into Γ measured by Trens and Denoyel23 (Γ)corrected =

(Γ)measured f

(74)

where f represents the ratio of the actual surface area of the solid, which is covered by PEG, to the total surface area of the solid. We estimate the correction factor so that the data of Trens and Denoyel23 matches with ours at higher values of the surface excess. The resulting plot is shown in Figure 10b. The value of f is 0.68, indicating that 32% of the surface area of the solid is not accessible to the polymer. The model fits the corrected data of Trens and Denoyel23 even up to low values of the surface excess. 4.3. Prediction of the Surface Excess Using the Estimated Values of χ*. The values of χ* estimated by regression of the calorimetry data have been used for the estimation of the surface excess of PEG on silica. These predictions are compared with the experimental data on PEGsilica in Figure 11. In general, the match is good. 4.4. Verification of the Postulate of Killman et al. Lastly, we verify the postulate of Killman et al.,22 who assert that the total heat of adsorption is directly proportional to the amount of the polymer in the train-form. Hence the heat of adsorption per unit surface excess ΔQΓI is proportional to the bound fraction, p (fraction of the total surface excess, in the form of trains). Since monomer adsorbs only in the form of the train, p = 1 for the monomer. Hence we can write ΔQ IΓ ΔQ mΓ

Figure 12. The heat of adsorption versus bound fraction; PEG20K, T = 298.15 K. The experimental data are shown by points: (■) PEG8K, (●) PEG20K, (▲) PEG100K. The solid line represents the linear fit to the experimental data.

ij A yz zz(φ* − ϕ b) ΔQ mΓ = RT jjjj z p j n pab zz pm k {

(76)

Here φpm * is the area fraction occupied by the monomer on the surface. Noting that Γb = np/A = φpm * /ab, eq 76 can be further modified to ΔQ mΓ

=p (75)

ij ϕpb yzz jj zz = RT jjj1 − * zzz j φpm k {

(77)

φpm * can be related to ϕbp by the Boltzmann weighting factor

where ΔQΓm represents the heat of adsorption of the monomer (note that both ΔQΓI and ΔQΓm are computed per basemol). Figure 12 is the plot of ΔQΓI versus the bound fraction. The plot is indeed a straight line, passing through the origin. The slope of the straight line should be equal to ΔQΓm. This yields the value of ΔQΓm = 2.03 kJ basemol−1. We have tried to reconcile this value with our theory as follows For the monomer, eq 14 combined with eq 15 for ΔQ̂ I can be simplified as follows. Since the monomer lies only in the surface phase, it has no contribution from the interphase. Hence,

* = ϕ b exp(−u p*/(RT )) φpm p

(78)

u*p is obtained by the following simplified form of eq 52 u p* RT

=−

ap aw

χ* (79)

In writing the above equation we assumed that the monomer retains its stiffness as a part of the Kuhn segment. N


Industrial & Engineering Chemistry Research Combining eq 77 with eq 79, we obtain ÄÅ É ÅÅ ij a p yzÑÑÑÑ Å Γ ΔQ m = RT ÅÅÅ1 − expjjj− χ * zzzÑÑÑ j a w zÑÑ ÅÅ k {ÑÖ ÅÇ

■ (80)

■

CONCLUSION Through this work, we have demonstrated that microcalorimetry is a sensitive technique to quantify the polymer− surface interaction. We have also developed a methodology for estimating the polymer−surface interaction parameter χ* from the heat of adsorption. Using this technique, it is possible to avoid the use of the displacer, which introduces inaccuracies in the measurements. A continuum model based on the self-consistent field theory has been used for estimating χ* through the heat of adsorption. The model has certain advantages over the lattice theory as pointed out in the test. However, it has also certain assumptions which are made to simplify the derivation and these would affect the estimate of χ*. For example, the surface phase concept assumes that the surface generates a discontinuous square well potential. Also, the thickness of the surface phase (which essentially consists of polymer segments in the train form and the adsorbed solvent molecules) is zero, so that we can express the concentrations in terms of area fractions instead of volume fractions. We have also assumed that the volume ratio of polymer segment to solvent is the same as the corresponding area ratio in the surface phase, but the present model can predict both the adsorbed amount of the polymer and also the train fraction of the polymer based on the value of χ* estimated using calorimetry. The model, therefore, passes in the consistency test. More stringent tests are however needed to bring out the scope and the limitation of the model. ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.8b04792.

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REFERENCES

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Substituting χ* = 0.512 and ap/aw = 3.27 into eq 80, we obtain ΔQΓm = 2.01 kJ basemol−1, which agrees well with the slope of the plot in Figure 12.

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The heat of adsorption estimated from Figures 1 and 2 (PDF)

AUTHOR INFORMATION

Corresponding Author

*Tel.: +91-22-27403115. Fax: +91-22-27403299. E-mail: lalaso. [email protected]. ORCID

Lalaso V. Mohite: 0000-0002-4188-2287 Jyoti Sahu: 0000-0003-0485-6298 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to thank (i) Unilever Industries Private Limited for providing the funding for the research and (ii) Prof. V. M. Naik from Department of Chemical Engineering, Indian Institute of Technology, Bombay, for valuable suggestions. O


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Quantification of Polymer-Surface Interaction using Microcalorimetry

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