The Effect of Temperature on the London Dispersive and Lewis Acid-Base Surface Energies of Polymethyl Methacrylate Adsorbed on Silica by Inverse Gas Chromatography

Abstract

Inverse gas chromatography at infinite dilution was used to determine the surface thermodynamic properties of silica particles and PMMA adsorbed on silica, and more particularly, to quantify the London dispersive energy ${\gamma }_s^d$, the Lewis acid ${\gamma }_s^{+}$, and base ${\gamma }_s^{-}$ polar surface energies of PMMA/silica composites as a function of the temperature and the recovery fraction $\theta$ of PMMA. The polar acid-base surface energy ${\gamma }_s^{AB}$ and the total surface energy of the different composites were then deduced as a function of the temperature. In this paper, the Hamieh thermal model was used to quantify the surface thermodynamic energy of polymethyl methacrylate (PMMA) adsorbed on silica particles at different recovery fractions. A comparison of the new results was carried out with those obtained by applying other molecular models of the surface areas of organic molecules adsorbed on the different solid substrates. An important deviation of these molecular models from the thermal model was proved. The determination of ${\gamma }_s^d$, ${\gamma }_s^{+}$, ${\gamma }_s^{-}$, and ${\gamma }_s^{AB}$ of PMMA in both the bulk and adsorbed phases showed an important non-linearity variation of these surface parameters as a function of the temperature. The presence of maxima in the curves of ${\gamma }_s^d(T)$ highlighted the second-order transition temperatures in PMMA showing beta-relaxation, glass transition, and liquid–liquid temperatures. These three transition temperatures depended on the adsorption rate of PMMA on silica. The proposed method gave a new relation between the recovery fraction of PMMA and its London dispersive energy, showing an important effect of the temperature on the surface energy parameters of the adsorption of PMMA on silica. A universal equation relating ${\gamma }_s^d(T,\theta )$ of the systems PMMA/silica to the recovery fraction and the temperature was proposed.

1. Introduction

The estimation of the London dispersive and acid-base surface energies of solids, oxides, and polymers [1,2,3,4,5,6,7,8,9,10,11,12,13,14] is of crucial importance in many industrial processes, such as adhesion, paints, coatings, corrosion, chemical reactions, adsorption, and catalysis. The most used technique to determine these surface parameters was inverse gas chromatography (IGC) at infinite dilution. The net retention time and volume of organic solvents adsorbed on solid surfaces, experimentally obtained by IGC, are directly correlated to the surface thermodynamic properties of solid materials, such as the London dispersive ${\gamma }_s^d(T)$ and polar ${\gamma }_s^p(T)$ components of the surface energy of solid particles as a function of the temperature. The interesting IGC technique used to characterize the surface and polar characteristics of materials was applied more particularly to determine the London dispersive energy of lignocellulosic fibers [15] based on works of Jacob et al. [16], Carvalho et al. [17], Chtourou et al. [18], Dorris and Gray [19], and Donnet et al. [20,21].

Gamble et al. [22] determined that the surface energy characteristics of a drug substance coated with two grades of silicon dioxide were studied by IGC using the Dorris and Gray approach. The IGC technique was used by Balard et al. [23] to determine the surface properties of milled graphites, by Bogillo et al. [24] to evaluate the surface-free energy components for heterogeneous solids, and by Das et al. [25,26] to study the surface energy distributions of lactose and pharmaceutical powders. The adsorption of n-alkanes at zero surface coverage on cellulose paper and wood fibers [19,27] and solid surface polarity [28] using IGC at infinite dilution were studied. Feeley et al. [29] studied the surface properties and flow characteristics of salbutamol sulphate, and the surface energy characteristics of micronized materials were determined [30]. The same technique was also used to measure the surface energy and high-energy sites for mixtures of crystalline and amorphous lactose [31,32]. Kołodziejek et al. [33] studied the relationship between surface properties determined by inverse gas chromatography and ibuprofen release from hybrid materials based on fumed silica, whereas Ho et al. [34,35,36] studied the surface energy heterogeneity of crystals with IGC. Many authors studied the surface energies of some polymers, such as tosylate functionalized poly(ethylene glycol) [37], polybutadiene/fillers [38], poly(2,2,3,3,3-pentafluoropropyl methacrylate) [39], and hyperbranched poly(ester amide) [40].

Several studies were interested in the adsorption of polymers on solid substrates or metallic oxides and their behaviors near the glass transition [41,42,43,44,45,46]. Other works were devoted to the segmental relaxation dynamics of poly(methyl methacrylate)/silica nanocomposites near and above the glass transition temperature by mechanical spectroscopy [47] and to the physical aging behavior under different adsorption conditions of PMMA chains [48,49,50,51,52].

In previous works [53,54], we studied the adsorption of PMMA on silica and alumina particles and the glass transition using inverse gas chromatography. However, classic chromatographic methods were used to determine the dispersive and polar-free energies of adsorbed organic molecules on the solid surfaces. In a recent study [55], the previous results were corrected using the new methodology based on the London dispersion interaction, and the new Lewis acid-base properties of PMMA/silica were obtained. The various dispersive, polar, and Lewis acid-base surface energies of the composites were not studied in the literature in light of the new thermal model.

The concept of the surface free energy ${\gamma }_s$ of material was first introduced by Fowkes [56] and resulted from a variety of intermolecular forces, such as the hydrogen bond, metallic bond, and London dispersion interaction [57,58]. The general expression of ${\gamma }_s$ can be given by

$${\gamma }_s={\gamma }_s^d+{\gamma }_s^p+{\gamma }_s^i+{\gamma }_s^c+{\gamma }_s^m$$

(1)

where ${\gamma }_s^d$ is the London dispersive component of the surface energy; the symbols p, i, c, and m are, respectively, the polar (due to Debye, Keesom, hydrogen, and acid-base interactions), ionic, covalent, and metallic components of the surface energy of the material. Relation 1 can be, in general, expressed as follows:

$${\gamma }_s={\gamma }_s^d+{\gamma }_s^{nd}$$

(2)

where ${\gamma }_s^{nd}$ includes all non-dispersive interaction forces. The non-dispersive component can be also symbolized by the total polar contribution ${\gamma }_s^p$ of the surface energy.

In this paper, we used the IGC technique [35,36,37,38,39,40,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83] at infinite dilution to determine the dispersive ${\gamma }_s^d(T)$ and polar ${\gamma }_s^p(T)$ components of the surface energy, as well as Lewis’s acid-base surface energies of poly(methyl methacrylate) (PMMA) adsorbed on silica at different recovery fractions as a function of the temperature using our new approach based on the Hamieh thermal model [59,60,61] that gave the variations of the surface area of adsorbed organic molecules as a function of the temperature.

2. IGC Methods and Materials

The experimental values of the net retention time $t_n$ and volume $V_n$ of the adsorbed organic solvents on PMMA and silica were obtained from the direct measurements using inverse gas chromatography (IGC) at infinite dilution [59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83]. The London dispersive energy was determined using the concept of Fowkes based on the geometric mean of the respective values of the London dispersive components of the surface energy of the non-polar molecule ${\gamma }_l^d$ and the solid substrate ${\gamma }_s^d$. The non-polar solvents generally used in the IGC technique were the n-alkanes (from n-pentane C5 to n-decane C10). The determination of $RTln{V}_n$ of n-alkanes adsorbed on PMMA/silica at different temperatures will lead to the values of ${\gamma }_s^d(T)$ using the following relation:

$$RTln{V}_n=2\mathcal{N}a{\left({\gamma }_l^d{\gamma }_s^d\right)}^{1/2}+\alpha (T)$$

(3)

where T is the absolute temperature of the chromatographic column containing the solid material, R is the perfect gas constant, $a$ is the surface area of an adsorbed molecule, $\mathcal{N}$ is the Avogadro number, and $\alpha (T)$ is a constant depending only on the temperature and the solid material.

Relation 1 was used to deduce the value of ${\gamma }_s^d$ by supposing both ${\gamma }_l^d$ and $a$ parameters as constants. Hamieh criticized the above approximation in several studies [59,60,61,81,82,83] by proving an important thermal effect on the surface area of organic molecules adsorbed on solid surfaces. We previously proposed different molecular models allowing for the determination of the surface areas of molecules [76], using Kiselev results, the two-dimensional Van der Waals (VDW) and Redlich–Kwong (R–K) equations, and the geometric, cylindrical, and spherical model. However, the most accurate model was based on the thermal model [59,60,61], which gave the expressions of the surface area $a\left(T\right)$ of organic molecules as a function of the temperature and those of ${\gamma }_l^d(T)$ relative to the different molecules.

A commercial Focus GC gas chromatograph apparatus (Sigma-Aldrich, St. Quentin Fallavier, France) equipped with a flame ionization detector (FID) was used in this study to obtain the net retention time and volume of different n-alkanes that were injected into a stainless-steel column (with a 2 mm inner diameter and a length of 20 cm) containing the solid particles. The temperature range varied from 30 °C to 200 °C. The same experimental procedure as those used in other studies [64,65,66,67,68] was applied to PMMA/silica. The used n-alkanes (pentane, hexane, heptane, octane, nonane) of a highly pure grade (99%) were purchased from a local chemical society.

The flow rate of the carrier gas (helium) was 20 mL/min, whereas the temperatures of the injector and detector were fixed at 200 °C. The infinite dilution of the probes was realized with the help of 1 µL Hamilton syringes by injecting very small quantities of the vapor solvent, satisfying the limit of detection of the FID of high sensitivity to practically realize the zero-surface coverage [84]. All columns containing PMMA and silica particles were preconditioned at 130 °C overnight to remove any adsorbed water molecules or other residual impurities.

PMMA and silica solid particles with different recovery fractions of PMMA (varying from 0 to 1.0) used in this work were the same solid materials previously characterized in other studies using other chromatographic methods [53,54].

3. Experimental Results

3.1. London Dispersive Surface Energy of Silica Particles

The values of $RTln{V}_n$ and $2\mathcal{N}a{\left({\gamma }_l^d\right)}^{1/2}$ of silica particles at different temperatures are given in Tables S1 and S2 by taking into account the thermal effect of the temperature on the surface area of organic solvents. Figure 1 gives the variations of ${\gamma }_s^d (T)$ of silica as a function of the temperature using the Hamieh thermal model.

Figure 1 shows that the monotonous decrease in the London dispersive surface energy of silica is not perfectly linear. The non-linear variation of ${\gamma }_s^d(T)$ can be approached by parabolic interpolation in this case. Equation (2) was obtained with an excellent regression coefficient (R² = 0.9994) in the studied domain of temperature as follows:

$${\gamma }_s^d\left(T\right)=2.8\times {10}^{-3}{T}^2-2.76T+691.29$$

(4)

Several molecular models of the surface area of n-alkanes were used: the geometrical model (based on the real surface area of molecules), the cylindrical molecular model (supposing a cylindrical form of molecule), the spherical model (supposing a spherical geometric form), Kiselev results, and the models using the two-dimensional Van der Waals and Redlich–Kwong equations transposed from three-dimensional space to two-dimensional space.

The comparison of this result with that obtained using other molecular models is shown in Figure 2.

The results presented in Figure 2 and

Table 1. Values of ${\gamma }_s^d \left(T\right) (\mathrm{i}\mathrm{n} \mathrm{m}\mathrm{J}/{\mathrm{m}}^2)$ of silica particles as a function of the temperature using the various models and the new Hamieh thermal model.

T (K)	Hamieh Thermal Model	Geometric	Cylindrical	Van Der Waals	Kiselev	Redlich–Kwong	Spherical
303.15	116.29	64.36	85.89	90.32	87.55	147.35	253.33
313.15	106.48	60.64	80.22	83.74	81.35	136.6	232.09
323.15	95.37	55.98	73.36	75.97	73.97	123.92	207.92
328.15	88.87	53.59	69.12	71.26	69.48	116.24	193.73
333.15	86.42	52.4	67.96	69.78	68.13	113.83	188.45
338.15	81.23	50.33	64.91	66.22	64.86	108.02	177.77
343.15	77.81	48.84	62.64	63.73	62.39	103.95	169.66
348.15	73.68	47.11	60.05	60.79	59.6	99.16	160.63
353.15	70.45	45.91	58.16	58.57	57.51	95.54	153.56
363.15	62.86	42.65	53.3	53.1	52.31	86.61	136.95
373.15	54.33	38.5	47.41	46.68	46.14	76.13	118.24
378.15	51.68	37.5	45.8	44.8	44.37	73.06	112.41
383.15	48.93	36.39	44.05	42.8	42.48	69.8	106.33
388.15	45.55	34.78	41.72	40.22	40.01	65.6	98.87
393.15	43.01	33.48	39.78	38.08	37.96	62.11	92.62
398.15	40.28	32.5	38.22	36.29	36.27	59.19	87.27
403.15	38.02	31.63	36.78	34.62	34.69	56.47	82.25
408.15	35.25	30.3	34.81	32.47	32.62	52.95	76.13
413.15	31.13	29.07	32.93	30.41	30.64	49.59	70.32
423.15	27.85	26.93	29.55	26.65	27.04	43.46	59.71
433.15	23.93	25.9	27.45	24.21	24.7	39.48	52.37
443.15	21.53	19.93	22.67	21.38	21.41	34.88	46.62
453.15	18.74	18.79	20.75	19.11	19.26	31.17	43.1
463.15	16.34	18.13	19.28	17.23	17.49	28.11	37.52
473.15	15.47	15.65	17.5	16.96	16.16	27.66	37.41

show the monotonous variations of ${\gamma }_s^d(T)$ of silica particles for all used molecular models with parabolic curves. However, an important deviation of all models was observed with respect to the Hamieh thermal model [59,60,61]. Indeed, the spherical model overestimated the values of the London dispersive surface energy of solid materials, whereas the geometric model underestimated the values of ${\gamma }_s^d(T)$, while the cylindrical, Kiselev, and Van der Waals models were the closest to that of the thermal model. The different molecular models did not take into consideration the thermal effect on the surface area. This incorrect assumption is the real cause of the erroneous values of ${\gamma }_s^d$ obtained by applying the different molecular models in comparison to those obtained by the thermal model.

This large deviation (Table 1) resulted from the fact that the effect of the temperature on the surface area of molecules was neglected in the above models. The thermal model gave different expressions of the variations of the surface area of organic molecules as a function of the temperature, whereas the other molecular models supposed all surface areas as constant. Therefore, only the results of the thermal model can be considered as accurate. The closest results to those of the thermal model were obtained by the Van der Waals model, perhaps because this model used the critical coordinates proper of each molecule. The Kiselev and cylindrical models gave similar results as those of Van der Waals with a deviation approaching 30%. The hypothesis that these above models used the cylindrical geometry of the various n-alkanes was a real hypothesis and took into account the adsorption position of such molecules but neglected, at the same time, the thermal effect. The maximum deviation was obtained with the spherical model due certainly to the use of the largest approximation considering a spherical geometry for n-alkanes, and these molecules exhibit generally linear forms.

3.2. Study of ${\gamma }_s^d(T)$ of PMMA and PMMA/Silica Composites

Figure 3 brings together the results of the London dispersive surface energy of PMMA particles obtained by the various applied molecular models compared to those of the thermal model. It can be observed that the results of the molecular models extremely deviate from those of the Hamieh thermal model. However, the general tendency of all drawn curves in Figure 3 and shown in

Table 2. Values of ${\gamma }_s^d\left(T\right) (\mathrm{i}\mathrm{n} \mathrm{m}\mathrm{J}/{\mathrm{m}}^2)$ of PMMA as a function of the temperature using the different models. The red bold values represent the coordinates of the maxima of ${\gamma }_s^d\left(T\right)$ indicating the presence of the transition temperatures of PMMA.

T(K)	Hamieh Thermal Model	Geometric	Cylindrical	Van Der Waals	Kiselev	Redlich–Kwong	Spherical
303.15	58.87	32.58	43.48	45.73	44.32	74.61	128.26
313.15	50.68	29.5	39.03	40.75	39.58	66.47	112.94
323.15	47.11	27.65	36.23	37.53	36.54	61.23	102.73
328.15	49.08	29.49	38.44	39.65	38.66	64.69	107.82
333.15	56.62	34.33	44.52	45.72	44.63	74.58	123.47
338.15	47.53	29.45	37.98	38.76	37.96	63.22	104.05
343.15	42.03	26.38	33.83	34.43	33.7	56.16	91.65
348.15	38.03	24.31	30.99	31.38	30.76	51.19	82.93
353.15	35.14	22.89	29	29.22	28.68	47.66	76.6
363.15	31.31	21.23	26.54	26.45	26.05	43.14	68.21
373.15	31.53	22.34	27.51	27.09	26.78	44.18	68.63
378.15	39.80	28.87	35.26	34.49	34.17	56.26	86.57
383.15	53.54	39.82	43.41	42.36	41.9	69.1	116.35
388.15	35.24	26.89	32.26	31.12	30.95	50.75	76.52
393.15	26.59	20.85	24.77	23.72	23.64	38.68	57.69
398.15	23.08	18.62	21.89	20.79	20.78	33.91	50.01
403.15	20.31	16.9	19.65	18.51	18.53	30.18	43.96
408.15	18.87	16.22	18.63	17.38	17.46	28.35	40.77
413.15	17.54	15.63	17.71	16.35	16.74	26.67	37.83
423.15	19.53	18.88	20.72	18.69	18.96	30.48	41.87
433.15	23.16	25.01	23.62	22.91	23.47	37.37	49.22
443.15	17.91	16.58	18.85	17.79	17.81	29.01	41.27
453.15	11.48	11.5	12.71	11.71	11.79	19.1	26.4
463.15	8.58	9.51	10.12	9.05	9.18	14.76	19.71
473.15	7.93	8.04	8.98	8.68	8.29	14.16	19.16

highlight the same values of the three maxima of temperatures. These three maxima showed the presence of three transition temperatures that were proved in other studies [53,54]. The beta-relaxation temperature, the glass transition, and the liquid–liquid temperatures were, respectively, obtained at 60 °C, 110 °C, and 160 °C, located on the different maxima of ${\gamma }_s^d(T)$, as shown in Figure 3 and Figure 4 and Table 2 (in bold dark red). The curves obtained by the thermal model and those of the cylindrical, Kiselev, geometric, and Van de Waals models are mixed up after a temperature of 400 K.

To better understand the effect of the various transition phenomena on the London dispersive surface energy, the one represented in Figure 4, the evolution of ${\gamma }_s^d \left(T\right)$ of PMMA as a function of the temperature using the thermal model and the corresponding curve of ${\gamma }_s^d \left(T\right)$ supposes that there is no transition temperature and shows the transition temperatures. The dashed line shows a parabolic monotonous decrease in ${\gamma }_s^d$ when the temperature increases. The equation of ${\gamma }_s^d(T)$ of PMMA without transition temperature is given by

$${\gamma }_s^d\left(T\right)=1.2\times {10}^{-3}{T}^2-1.26T+324.00$$

(5)

A comparison with Equation (4) relative to silica particles showed that the values of ${\gamma }_s^d \left(T\right)$ of silica are approximately double those of PMMA without transition phenomenon. On the other hand, the comparison between the two curves in Figure 4 showed that before every transition temperature, the London dispersive surface energy of PMMA increased until it reached its maximum value relative to the transition temperature with unstable equilibrium and again decreased to join the dashed line. The variations of ${\gamma }_s^d\left(T\right)$ around the transition temperature are due to the intense motion and fluctuations of the chain groups of the polymer. Table 2 clarifies the values of ${\gamma }_s^d\left(T\right)$ when the temperature increased.

The highest non-linearity of the curves ${\gamma }_s^d (T)$ of PMMA was observed in all molecular models in the temperature interval [300 K; 500 K]. This non-linearity is certainly due to the presence of transition phenomena in PMMA submitted to the surface group changes when the temperature increases. All molecular models largely deviate from the thermal model. The deviation reached 300% in many cases. However, the results obtained by those models can be only considered qualitative, whereas the thermal model gave the most accurate results.

The results relative to the adsorption of PMMA on silica particles are presented in Figure 4 and

Table 3. Values of ${\gamma }_s^d (T)$ of PMMA adsorbed on silica particles as a function of the temperature at different recovery fractions from $\theta =0.31$ to $\theta =1.0$ using the various molecular models. The bold dark red figures are relative to the transition temperatures.

$\mathbf{Adsorption} \mathbf{of} \mathbf{PMMA} \mathbf{on} \mathbf{Silica} \mathbf{for} \mathit{\theta }=0.31$
T (K)	Hamieh Thermal Model	Geometric	Cylindrical	Van Der Waals	Kiselev	Redlich–Kwong	Spherical
303.15	104.37	57.76	77.09	81.07	78.57	132.24	227.37
313.15	91.30	51.99	68.78	71.80	69.75	117.13	199.00
323.15	81.98	48.12	63.06	65.31	63.59	106.54	178.75
328.15	78.95	47.09	61.39	63.31	61.73	103.28	172.15
333.15	80.44	48.78	63.26	64.95	63.41	105.95	175.41
338.15	81.52	50.51	65.15	66.46	65.09	108.41	178.40
343.15	83.10	52.17	66.90	68.05	66.63	111.01	181.19
348.15	73.51	47.00	59.91	60.65	59.46	98.93	160.25
353.15	64.73	42.18	53.43	53.82	52.84	87.78	141.09
363.15	53.21	36.09	45.11	44.94	44.27	73.31	115.91
373.15	43.45	30.79	37.91	37.33	36.90	60.88	94.56
378.15	41.08	29.80	36.40	35.60	35.27	58.07	89.35
383.15	39.48	29.35	35.54	34.53	34.27	56.33	85.81
388.15	40.75	31.10	37.30	35.98	35.79	58.69	88.48
393.15	45.69	35.85	42.56	40.74	40.62	66.45	99.10
398.15	50.97	41.15	48.37	46.28	45.89	74.90	110.42
403.15	43.24	36.00	41.84	39.38	39.46	64.23	93.55
408.15	35.25	30.10	34.52	32.17	32.33	52.47	75.41
413.15	28.80	25.66	29.07	26.84	27.04	43.78	62.08
423.15	22.94	22.18	24.34	21.95	22.27	35.80	49.18
433.15	23.99	21.38	22.44	19.60	20.07	31.96	42.09
443.15	27.13	25.13	28.57	29.95	26.99	43.95	62.52
453.15	19.05	19.10	21.09	19.42	19.57	31.68	43.80
463.15	14.48	16.06	17.08	15.27	15.50	24.91	33.25
473.15	13.85	14.04	13.03	10.36	11.03	16.89	20.69
Adsorption of PMMA on Silica for $\mathit{\theta }=0.54$
T (K)	Hamieh Thermal Model	Geometric	Cylindrical	Van Der Waals	Kiselev	Redlich–Kwong	Spherical
303.15	95.71	53.22	70.69	74.34	72.05	121.26	208.47
313.15	84.53	48.13	63.67	66.47	64.57	108.43	184.23
323.15	74.85	43.93	57.57	59.62	58.06	97.26	163.19
328.15	72.05	42.97	56.03	57.78	56.34	94.26	157.12
333.15	73.29	44.44	57.63	59.17	57.77	96.53	159.81
338.15	76.92	47.65	61.43	62.71	61.42	102.29	168.35
343.15	71.19	44.68	57.31	58.31	57.08	95.12	155.23
348.15	65.49	41.87	53.37	54.03	52.97	88.14	142.77
353.15	58.91	38.09	48.43	49.22	48.08	80.29	129.15
363.15	48.96	33.21	41.51	41.36	40.74	67.46	106.65
373.15	38.16	27.04	33.3	32.78	32.41	53.47	83.06
378.15	36.12	26.2	32	31.31	31.01	51.06	78.57
383.15	38.33	28.5	34.51	33.53	33.28	54.69	83.32
388.15	42.68	32.58	39.08	37.7	37.5	61.49	92.7
393.15	52.06	40.83	48.5	46.42	46.28	75.72	112.92
398.15	46.88	37.86	44.5	42.24	42.22	68.89	101.55
403.15	40.96	34.09	39.63	37.3	37.37	60.84	88.61
408.15	31.95	27.28	31.29	29.16	29.31	47.56	68.36
413.15	24.69	22.04	24.95	23.01	23.19	37.52	53.18
423.15	20.39	19.71	21.63	19.51	19.79	31.82	43.71
433.15	26.55	23.63	24.79	21.66	22.18	35.31	46.51
443.15	17.74	16.42	18.67	17.61	17.64	28.73	40.87
453.15	14.20	14.35	15.85	14.61	14.71	23.82	32.94
463.15	12.12	13.44	14.29	12.78	12.97	20.84	27.82
473.15	11.41	11.57	12.92	12.85	11.93	20.38	27.57
Adsorption of PMMA on Silica for $\mathit{\theta }=0.83$
T (K)	Hamieh Thermal Model	Geometric	Cylindrical	Van Der Waals	Kiselev	Redlich–Kwong	Spherical
303.15	83.96	46.46	62.01	65.21	63.21	106.39	182.9
313.15	76.57	43.6	57.67	60.22	58.49	98.23	166.91
323.15	66.94	39.29	51.49	53.33	51.93	87	145.96
328.15	69.57	41.49	54.1	55.79	54.4	91.02	151.71
333.15	74.62	45.24	58.68	60.25	58.82	98.29	162.72
338.15	69.65	43.16	55.67	56.79	55.62	92.64	152.46
343.15	64.16	40.28	51.65	52.55	51.44	85.72	139.91
348.15	58.34	37.3	47.55	48.14	47.19	78.53	127.2
353.15	52.02	33.95	42.98	43.21	42.47	70.48	113.27
363.15	43.21	29.31	36.63	36.5	35.96	59.54	94.14
373.15	33.20	23.52	28.97	28.52	28.19	46.52	72.26
378.15	30.44	22.08	26.97	26.38	26.13	43.04	66.22
383.15	35.11	26.1	31.61	30.71	30.48	50.09	76.31
388.15	50.63	38.41	46.07	44.44	44.2	72.48	109.28
393.15	40.82	32.01	38.03	36.4	36.29	59.37	88.54
398.15	34.59	27.92	32.83	31.17	31.15	50.84	74.96
403.15	29.88	24.86	28.9	27.21	27.26	44.38	64.64
408.15	24.15	20.61	23.64	22.04	22.15	35.94	51.66
413.15	19.32	17.21	19.5	18.01	18.14	29.37	41.65
423.15	16.27	15.72	17.26	15.57	15.8	25.39	34.89
433.15	27.30	24.3	25.49	22.26	22.8	36.31	47.82
443.15	17.37	16.08	18.28	17.25	17.27	28.14	40.03
453.15	12.73	12.76	14.1	12.99	13.08	21.19	29.29
463.15	9.74	10.79	11.48	10.27	10.42	16.75	22.36
473.15	9.17	9.30	10.39	10.04	9.59	16.38	22.16
Adsorption of PMMA on Silica for $\mathit{\theta }=0.98$
T (K)	Hamieh Thermal Model	Geometric	Cylindrical	Van Der Waals	Kiselev	Redlich–Kwong	Spherical
303.15	80.49	44.54	59.44	62.52	60.6	101.99	175.35
313.15	72.76	41.43	54.81	57.22	55.58	93.35	158.6
323.15	63.27	37.13	48.66	50.4	49.07	82.22	137.94
328.15	65.85	39.28	51.21	52.81	51.49	86.15	143.6
333.15	70.99	43.04	55.83	57.32	55.96	93.51	154.82
338.15	64.89	40.21	51.86	52.91	51.82	86.31	142.04
343.15	59.74	37.5	48.09	48.93	47.9	79.81	130.25
348.15	54.14	34.5	44.12	44.68	43.79	72.87	117.65
353.15	49.63	32.34	40.97	41.27	40.52	67.32	108.2
363.15	40.84	27.7	34.62	34.5	33.98	56.27	88.97
373.15	31.81	22.54	27.76	27.33	27.02	44.58	69.25
378.15	32.08	23.19	28.37	27.82	27.54	45.38	69.63
383.15	36.69	27.28	33.04	32.1	31.86	52.36	79.76
388.15	53.25	40.64	48.75	47.02	46.77	76.68	115.61
393.15	46.15	39.19	43.00	41.16	41.03	67.13	100.11
398.15	40.22	32.46	38.16	36.24	36.21	59.1	87.13
403.15	29.26	24.35	28.31	26.65	26.7	43.47	63.32
408.15	22.52	19.22	22.05	20.55	20.66	33.52	48.18
413.15	18.87	16.79	19.04	17.59	17.72	28.7	40.63
423.15	16.30	15.75	17.29	15.6	15.82	25.44	34.96
433.15	27.59	24.55	25.75	22.49	23.04	36.68	48.31
443.15	15.59	14.42	16.41	15.48	15.5	25.25	35.92
453.15	10.59	10.61	11.72	10.8	10.88	17.62	24.36
463.15	8.16	9.04	9.62	8.6	8.73	14.03	18.74
473.15	7.66	7.77	8.68	8.39	8.01	13.68	18.51
Adsorption of PMMA on Silica for $\mathit{\theta }=1.0$
T(K)	Hamieh Thermal Model	Geometric	Cylindrical	Van Der Waals	Kiselev	Redlich–Kwong	Spherical
303.15	75.92	42.01	56.07	58.97	57.16	96.2	165.4
313.15	67.65	38.52	50.96	53.21	51.68	86.8	147.47
323.15	58.85	34.54	45.27	46.89	45.65	76.4	128.33
328.15	61.70	36.8	47.98	49.48	48.24	80.71	134.54
333.15	67.28	40.79	52.91	54.33	53.04	88.62	146.73
338.15	61.50	38.11	49.15	50.15	49.12	81.8	134.63
343.15	56.07	35.19	45.14	45.93	44.96	74.92	122.27
348.15	50.99	32.6	41.56	42.08	41.25	68.64	111.18
353.15	46.12	30.05	38.07	38.35	37.65	62.55	100.54
363.15	37.96	25.75	32.18	32.07	31.59	52.31	82.71
373.15	28.47	20.17	24.84	24.46	24.18	39.9	61.97
378.15	26.94	19.54	23.87	23.36	23.13	38.09	58.61
383.15	40.51	30.1	36.45	35.41	35.15	57.76	87.99
388.15	55.89	42.66	51.17	49.35	49.09	80.5	121.36
393.15	47.51	37.26	44.26	42.37	42.24	69.1	103.05
398.15	39.16	31.61	37.16	35.29	35.26	57.55	84.85
403.15	28.98	24.11	28.03	18.92	26.44	43.04	62.69
408.15	22.77	19.43	22.29	20.78	20.88	33.88	48.71
413.15	18.42	16.41	18.59	17.17	17.3	28	39.71
423.15	15.79	15.26	16.75	15.11	15.33	24.65	33.87
433.15	25.97	24.38	25.61	22.38	22.93	36.5	48.15
443.15	15.28	14.14	16.09	15.18	15.2	24.76	35.23
453.15	10.19	10.21	11.28	10.4	10.47	16.96	23.44
463.15	7.71	8.58	9.11	8.13	8.26	13.26	17.7
473.15	7.39	7.49	8.37	8.09	7.72	13.19	17.85

at different recovery fractions varying from $\theta =0$ (the case of pure silica particles) to $\theta =1.0$ (representing a monolayer of PMMA on silica). The results in Figure 5 and Table 3 clearly show the large difference in the values of the London dispersive surface energy of the PMMA/silica composites when the temperature and the recovery fraction changed. Three interesting results were detected as follows:

-

The London dispersive surface energy decreases when the recovery fraction fractions of PMMA adsorbed on silica increases whatever the used molecular model;

-

The adsorption of PMMA on silica strongly affects the physicochemical properties of PMMA relative to its bulk phase. The transition temperatures observed in PMMA alone change with the recovery fraction;

-

When approaching the monolayer of adsorption of PMMA on silica particles, there is a spontaneous decreasing evolution of the London dispersive surface energy from silica particles (Table 1) to PMMA in the bulk phase (Table 2), and when passing through the different recovery fractions from $\theta =0.31$ to $\theta =1.0$ (Table 3 and Figure 5), the behavior of the PMMA monolayer becomes closer to PMMA in its bulk phase.

The results given in Figure 5 and Table 3 give the various transition temperatures of PMMA adsorbed on silica particles at different recovery fractions. The obtained values are presented in Table S3 and Figure 6. It was observed that the beta-relaxation temperature linearly decreased from $T_{\beta }=343.15 \mathrm{K}$ to stabilize at $T_{\beta }=333.15 \mathrm{K}$ for $\theta =0.83$, and the same value was observed for PMMA alone, whereas the glass transition decreased from $T_g=398.15 \mathrm{K}$ to $T_g=383.15 \mathrm{K}$ for $\theta =0.83$, and then it increased to $T_g=388.15 \mathrm{K}$ to reach the value $T_g=383.15 \mathrm{K}$ for PMMA. However, the liquid–liquid transition temperature oscillated between $T_{l-l}=443.15 \mathrm{K}$ and $T_{l-l}=433.15 \mathrm{K}$. Therefore, there is an important effect of the adsorption of PMMA on the London dispersive energy of the PMMA/silica systems and the transition temperatures. Those results were confirmed for all molecular models.

Furthermore, the London dispersive surface energy of the PMMA/silica composites at the different transition temperatures strongly varied as a function of the recovery fraction. These variations of ${\gamma }_s^d$ are clearly shown in

Table 4. Values of the London dispersive surface energy ${\gamma }_s^d (\mathrm{m}\mathrm{J}/{\mathrm{m}}^2)$ of PMMA/silica composites at the transition temperatures as a function of the recovery fraction.

$\mathbf{Recovery} \mathbf{Fraction} \mathbf{of} \mathbf{PMMA}/\mathbf{Silica} \mathit{\theta }$	Beta-Relaxation $\mathbf{Temperature} {\mathit{T}}_{\mathit{\beta }}$	Glass Transition $\mathbf{Temperature} {\mathit{T}}_{\mathit{g}}$	Liquid–Liquid $\mathbf{Temperature} {\mathit{T}}_{\mathit{l}-\mathit{l}}$
0.31	83.10	50.97	27.13
0.54	76.92	52.06	26.55
0.83	74.62	50.63	17.37
0.98	70.99	53.26	15.59
1.0	67.28	55.89	25.97
PMMA	56.62	53.54	23.16

. A decrease in the London dispersive energy was observed before reaching the monolayer in the cases of beta-relaxation and liquid–liquid transitions, whereas a small change was found in the case of the glass transition. Table 4 also shows that after the monolayer, the different values of ${\gamma }_s^d$ approached those of PMMA.

In fact, the above observations can be generalized for all temperatures and recovery fractions, as shown in Figure 7. The curves plotted in Figure 7a show the shift of the maxima of the various transition temperatures when the recovery fraction of PMMA/silica varied. Furthermore, the variations of ${\gamma }_s^d(\theta )$ were perfectly linear for all temperatures far from the transition temperatures, as shown in Figure 7b. The results show a decrease in the London dispersive surface energy of PMMA/silica when the temperature and/or the recovery increased. The different equations of ${\gamma }_s^d(\theta )$ are given in

Table 5. Equation of London dispersive surface energy ${\gamma }_s^d(\theta )$ of PMMA/silica composites at different temperatures far from the transition temperatures with the linear regression coefficients and the values of (−$d{\gamma }_s^d/d\theta$) and ${\gamma }_s^d\left(0\right)$.

T (K)	$\mathbf{Equation} \mathbf{of} {\mathit{\gamma }}_{\mathit{s}}^{\mathit{d}}(\mathit{\theta })$	R2	$-\mathit{d}{\mathit{\gamma }}_{\mathit{s}}^{\mathit{d}}/\mathit{d}\mathit{\theta }$	${\mathit{\gamma }}_{\mathit{s}}^{\mathit{d}}(0)$
303.15	${\gamma }_s^d(\theta )$ = −38.63 $\theta$ + 116.32	0.994	38.63	116.32
313.15	${\gamma }_s^d\left(\theta \right)$= −34.92 $\theta$ + 104.49	0.9779	34.92	104.49
323.15	${\gamma }_s^d(\theta )$ = −33.44 $\theta$ + 93.92	0.985	33.44	93.92
328.15	${\gamma }_s^d(\theta )$ = −23.85 $\theta$ + 87.36	0.9569	23.85	87.36
348.15	${\gamma }_s^d\left(\theta \right)$= −23.40 $\theta$ + 76.95	0.9293	23.40	76.95
353.15	${\gamma }_s^d(\theta )$ = −23.26 $\theta$ + 71.15	0.9865	23.26	71.15
363.15	${\gamma }_s^d(\theta )$ = −22.75 $\theta$ + 61.70	0.9836	22.75	61.70
373.15	${\gamma }_s^d(\theta )$ = −23.19 $\theta$ + 52.36	0.9658	23.19	52.36
413.15	${\gamma }_s^d(\theta )$ = −13.59 $\theta$ + 31.82	0.9765	13.59	31.82
423.15	${\gamma }_s^d(\theta )$ = −11.84 $\theta$ + 27.13	0.9751	11.84	27.13
463.15	${\gamma }_s^d(\theta )$ = −8.69 $\theta$ + 16.72	0.9918	8.69	16.72
473.15	${\gamma }_s^d(\theta )$ = −8.27 $\theta$ + 15.87	0.9904	8.27	15.87

. The linearity is shown by the good linear regression coefficients shown in Table 5.

Table 5 led to the following thermodynamic relations of (−$\frac{d{\gamma }_s^d}{d\theta }(T)$) and ${\gamma }_s^d(T, 0)$ as a function of the temperature as follows:

$$\left\{\begin{array}{c}-\frac{d{\gamma }_s^d}{d\theta }(T)=8\times {10}^{-4}{ T}^2-0.759 T+197.04; R^2=0.9453\\ {\gamma }_s^d\left(T,0\right)=2.8\times {10}^{-3}{ T}^2-2.77 T+696.64; R^2=0.9972\end{array}\right.$$

(6)

Relation 6 then gives a universal equation of the following form:

$${\gamma }_s^d\left(T, \theta \right)=\frac{d{\gamma }_s^d}{d\theta }\left(T\right) \theta +{\gamma }_s^d\left(T, 0\right)$$

(7)

Equation (7) is the general form of the dependence of the London dispersive surface energy of the PMMA/silica composites on the temperature and the recovery fraction $\theta$. The expression of ${\gamma }_s^d\left(T, 0\right)$ given in Relation 3 gave the same equation as that obtained with the silica particles (for $\theta =0$). The obtained linearity of ${\gamma }_s^d$ as a function of $\theta$ is not realized with the temperature. Indeed, the second-degree equation of (−$\frac{d{\gamma }_s^d}{d\theta }(T)$) and ${\gamma }_s^d(T, 0)$ versus the temperature gave a non-linearity dependence of ${\gamma }_s^d\left(T\right)$.

The Equation (7) can be written for a constant temperature as follows:

$$\theta \frac{d{\gamma }_s^d}{d\theta }-{\gamma }_s^d\left( \theta \right)+{\gamma }_s^d\left( 0\right)=0$$

(8)

Equation (8) is then an ordinary first-order differential equation of ${\gamma }_s^d\left( \theta \right)$. It can be easily integrated. ${\gamma }_s^d\left( \theta \right)$ is, therefore, given at any temperature T by Equation (13) as follows:

$${\gamma }_s^d\left( \theta \right)=\left({\gamma }_s^d\left( \theta =1\right)-{\gamma }_s^d\left( \theta =0\right)\right) \theta +{\gamma }_s^d\left( 0\right)$$

(9)

where ${\gamma }_s^d\left( 0\right)$ and ${\gamma }_s^d\left( 1\right)$ represent the respective values of the London dispersive energy of silica $\left( \theta =0\right)$ and PMMA/silica for $\theta =1$ at a fixed temperature T.

The results obtained from the integrated theoretical Equation (9) can be compared to the experimental results given in Table 5. As examples, we show in

Table 6. Theoretical results of ${\gamma }_s^d(\theta )$ of PMMA/silica composites at three temperatures with the values of (−$d{\gamma }_s^d/d\theta$) and ${\gamma }_s^d\left(0\right)$.

T (K)	$\mathbf{Equation} \mathbf{of} {\mathit{\gamma }}_{\mathit{s}}^{\mathit{d}}(\mathit{\theta })$	$-\mathit{d}{\mathit{\gamma }}_{\mathit{s}}^{\mathit{d}}/\mathit{d}\mathit{\theta }$	${\mathit{\gamma }}_{\mathit{s}}^{\mathit{d}}(0)$
303.15	${\gamma }_s^d(\theta )$ = −40.37 $\theta$ + 116.29	40.37	116.29
363.15	${\gamma }_s^d\left(\theta \right)$= −37.96 $\theta$ + 62.86	37.96	62.86
473.15	${\gamma }_s^d(\theta )$ = −8.08 $\theta$ + 15.47	8.08	15.47

the theoretical results at three temperatures of 303.15 K, 363.15 K, and 473.15 K.

The comparison between the theoretical results given in Table 6 with the experimental results given in Table 5 show excellent agreement. The difference between the two results did not exceed 2%.

3.3. Lewis’s Acid-Base Polar Surface Energies of PMMA/Silica

Van Oss et al.’s method [85] was used to determine the Lewis acid ${\gamma }_s^{+}$ and base ${\gamma }_s^{-}$ surface energies of PMMA/silica knowing the Lewis acid ${\gamma }_l^{+}$ and base ${\gamma }_l^{-}$ surface energies of the used solvents. Van Oss et al. [85] used two monopolar solvents, namely, ethyl acetate (EA) and dichloromethane (DCM), which are usually characterized by

$$\left\{\begin{array}{c}{\gamma }_{DCM}^{+}=5.2 \mathrm{m}\mathrm{J}/{\mathrm{m}}^2 , {\gamma }_{DCM}^{-}=0\\ {\gamma }_{EA}^{+}=0 , {\gamma }_{EA}^{-}=19.2 \mathrm{m}\mathrm{J}/{\mathrm{m}}^2\end{array}\right.$$

(10)

In previous studies [53,54], the polar-free energy $∆G_a^{sp}\left(T\right)$ of the different PMMA/silica composites was determined as a function of the temperature with various recovery fractions. Knowing that, $∆G_a^{sp}\left(T\right)$ can be expressed as follows:

$$∆G_a^{sp}\left(T\right)=2\mathcal{N}a\left(\sqrt{{\gamma }_l^{-}{\gamma }_s^{+}}+\sqrt{{\gamma }_l^{+}{\gamma }_s^{-}}\right)$$

(11)

The Lewis acid and base surface energies of the various PMMA/silica composites were determined with the help of the following equations:

$$\left\{\begin{array}{c}\begin{array}{c}{\gamma }_s^{+}=\frac{{\left[∆G_a^{sp}\left(T\right)\left(EA\right)\right]}^2}{{4\mathcal{N}}^2{\left[a\left(EA\right)\right]}^2{\gamma }_{EA}^{-}} \end{array} \\ {\gamma }_s^{-}=\frac{{\left[∆G_a^{sp}\left(T\right)\left(DCM\right)\right]}^2}{{4\mathcal{N}}^2{\left[a\left(DCM\right)\right]}^2{\gamma }_{DCM}^{+}} \end{array}\right.$$

(12)

using the values of the surface area of polar molecules obtained from the Hamieh thermal model. The obtained values of $∆G_a^{sp}\left(T\right)$ of EA and DCM adsorbed on the different solid surfaces are given in Tables S4 and S5. The polar (or acid-base) surface energy ${\gamma }_s^{AB}$ and the total surface energy of different PMMA/silica composites were obtained in Equation (13).

$$\left\{\begin{array}{c}{\gamma }_s^{AB}=2\sqrt{{\gamma }_s^{+}{\gamma }_s^{-}}\\ {\gamma }_s^{tot.}={{\gamma }_s^d+\gamma }_s^{AB}\end{array}\right.$$

(13)

The results are presented in Table S6 and Figure 8. All surface energy parameters of the various PMMA/silica varied as a function of the temperature, as shown in Figure 8.

Several conclusions can be deduced as follows:

-

All curves of the acid-base polar surface energies decreased against the temperature, except at the transition temperature where higher values of these parameters were observed for the different recovery fractions.

-

The acid polar surface energy of silica did not present any appreciated variation when the recovery fraction of PMMA increased. Globally, the acidity of silica particles is not very affected by the adsorbed amount of PMMA.

-

An important change in the value of the base polar surface energy ${\gamma }_s^{-}(T)$ was observed when the recovery fraction of PMMA varied. It seems that a maximum of ${\gamma }_s^{-}(T)$ was obtained in the case of a monolayer ($\theta =1.0)$ of PMMA adsorbed on silica particles, and the acid-base polar surface energy ${\gamma }_s^{AB}$ was the highest in this later case. For this value of $\theta =1.0$, a maximum of ${\gamma }_s^{tot.}(T)$ was also observed. In the case of the monolayer of adsorption of PMMA on silica, it can be deduced that the polar surface energy was higher than those of silica and PMMA taken separately, and it approximately approached the summation of these two extreme cases of $\theta =0 (Silica)$ and PMMA in the bulk phase.

-

Figure 8 shows all curves of the Lewis acid-base surface energies as a function of the temperature and gave similar variations as those of ${\gamma }_s^d(T)$. The behavior of these Lewis surface parameters showed an increase before every transition temperature until reaching their maximum values at the transition temperature directly followed by a decrease in the Lewis surface energies. These variations of neighboring transition temperatures are due to the fluctuations and relaxation of the chain groups of PMMA as a function of the temperature.

4. Conclusions

The adsorption of PMMA on silica particles at different recovery fractions was studied as a function of the temperature using inverse gas chromatography (IGC) at infinite dilution. The IGC technique gave the experimental values of $RTln{V}_n$ of n-alkanes adsorbed on the different PMMA/silica composites. Several molecular models were used and compared to the recent Hamieh thermal model to determine the variations of the London dispersive surface energy of solid particles as a function of the temperature and the recovery fraction of PMMA adsorbed on silica. The determination of ${\gamma }_s^d(T)$ of silica particles using the thermal model showed parabolic variations with an excellent regression coefficient approaching 1.000. The other molecular models gave the same type of variations of ${\gamma }_s^d(T)$ but with important deviation from the accurate thermal model, taking into account the effect of the temperature on the surface area of organic molecules. The curves of ${\gamma }_s^d(T)$ of the PMMA/silica systems confirmed the presence of three maxima, characterizing the three transition temperatures of the beta-relaxation, the glass transition, and the liquid–liquid transition. An important effect of the recovery fraction of PMMA on the values of the transition temperatures was highlighted in this study.

A universal equation of ${\gamma }_s^d\left(T, \theta \right)$ of the various PMMA/silica composites was given as a function of the two variables, the temperature $T$, and the recovery fraction $\theta$. The determination of the couple $\left(T, \theta \right)$ led to obtaining the value of ${\gamma }_s^d$ and the full thermodynamic determination of the PMMA/silica system.

The Lewis acid ${\gamma }_s^{+}$ and base ${\gamma }_s^{-}$ polar surface energies of PMMA/silica composites were determined using the thermal model as a function of the temperature and the recovery fraction. The polar acid-base surface energy ${\gamma }_s^{AB}$ and the total surface energy of the different composites were then deduced. It was shown that ${\gamma }_s^{+}$ of silica was not very affected by the adsorption of PMMA on silica, whereas ${\gamma }_s^{-}$ presented large variations when the recovery fraction varied. A maximum of basic and total polar surface energies was reached in the case of the monolayer of PMMA. The total polar surface energy of PMMA/silica for a monolayer ($\theta =1.0$) was proved to be approximately equivalent, out of the transition temperatures, to the summation of those obtained with silica and PMMA solid surfaces.