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Biorheology 40 (2003) 431–439 IOS Press

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The influence of humidity on the viscoelastic behaviour of human hair P. Zuidema a , L.E. Govaert a , F.P.T. Baaijens a , P.A.J. Ackermans b,∗ and S. Asvadi b a Materials

Technology (MaTe), Dept. Mechanical Engineering, Eindhoven University of Technology, P.O.box 513, NL-5600 MB, Eindhoven, The Netherlands b Philips Research, Personal Care Institute, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands Received 15 July 2002 Accepted in revised form 7 November 2002 Abstract. The possibility of curling hair is attributed to the fact that the mechanical properties of hair fibres depend on time, temperature and humidity. In this study the dependence of the viscoelastic behaviour of human hair fibres on humidity is characterised in a bending deformation. An experimental set-up was used to perform bending relaxation measurements at different humidities. The relaxation data were fitted using a so-called “stretched exponential equation”. The humidity dependence was incorporated by applying time–humidity superposition. Also, the influence of humidity on the initial E-modulus was found. The influence of humidity on the behaviour of human hair fibres could be modelled by using a general characterisation of the behaviour of human hair fibres. The general characterisation is used to predict the recovery in a human hair fibre after curling. The predicted recovery appeared to agree reasonably well with experimentally determined values. Keywords: Human hair, viscoelasticity, recovery, humidity dependence

1. Introduction Experimental evidence reveals that both wetting and heating hair make it easier to shape. Previous studies have demonstrated that human hair shows mechanical behaviour that can be described as viscoelastic [8,10,13]. The origin of this behaviour lies in the fact, that the hydrogen-bonds within a hair fibre are easily broken by stretching or bending in combination of heating and/or wetting, while the disulphide bonds remain unbroken. This molecular mechanism is similar to what has been found for wool fibres [10]. During bending under constant curvature, the continuous breaking and building of hydrogen-bond cross-links relieves the internal stresses in the molecular assembly, which leads to the lowering of the tension within the hair fibre. This behaviour has strong similarities with that of other cross-linked polymers. The disulphide bonds form the cross-links, while the hydrogen bonds are considered as secondary temporary bonds. During curling, a hair fibre is forced into a desired shape. Due to viscoelastic effects internal stresses will decrease (stress relaxation), which results in a remaining deformation (e.g., a curl) after release. The remaining deformation is therefore strongly determined by the amount of relaxation. In other words, the higher the relaxation, the better the curl is maintained. Under constant curvature, the magnitude of this remaining deformation (“level of set”), is mainly determined by strain level, loading time and * Address for correspondence: P.A.J. Ackermans, Philips Research, Personal Care Institute, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands. Fax: +31 40 2744288; E-mail: [email protected].

0006-355X/03/$8.00  2003 – IOS Press. All rights reserved

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environmental conditions (temperature, humidity). Based on the same principle of stress relaxation, the hair will always recover to its original shape. The time-dependent behaviour of human hair depends on the temperature and the relative water content, for which usually the regain (Rg ) is employed. This regain (Rg ) is defined as the ratio of the mass of absorbed water and the mass of the dry hair. By applying high temperature and/or moisture the time to reach equilibrium can be shortened. The aim of this study is to determine the benefits of viscoelastic modelling for describing the curling process in human hair. In particular, the role of humidity dependence is investigated. Moreover, the natural drying behavior of hair, as it occurs during curling, is incorporated in the modelling. The increased relaxation rate due to a higher temperature can be described by the principle of time– temperature equivalence [9]. In a similar reduced time approach, also the effects of humidity (regain Rg ) [4–6] and progressive aging [11] can be included. It is known that a change in regain will have the same effect on the relaxation rate of keratin fibres as a change in temperature [4–6,8,13]. In some of these studies [4–6], it was shown that physical aging also has a significant effect on the viscoelastic response of the fiber. Attempts to model this influence, in changing (transient) environmental conditions, are unknown to the authors. It is well known that aging effects, in the linear viscoelastic region, can successfully be modelled using a reduced time approach if the environmental conditions are constant [11]. However, as indicated in the work of Struik [11], this approach fails in situations where temperature (or in our case humidity) transients are applied. Therefore, the influence of aging is not included in the present study. To assure similar influence of physical aging all specimens, used for the viscoelastic characterisation, were pre-conditioned in the same way. The viscoelastic functions obtained in this manner are subsequently applied to predict curl-recovery in a transient-humidity situation (as in a realistic curling process) where a wet hair is allowed to dry (and recover) at room temperature. In previous studies, besides an effect on the relaxation rate, also an influence on the initial Young’smodulus (E) was found [13]. Qualitatively, it is observed that the initial Young’s-modulus will decrease at higher regain. 2. Experimental To measure the time dependence of the modulus of elasticity a three-point bending test was used. The modulus of elasticity of hair could be determined by supporting the sample at both ends and measuring the load that resulted from the enforced deformation at the centre. This was done by using a highly sensitive balance of a Perkin-Elmer TGS-2 Thermogravimetric System. This system used a servomotor to compensate the difference between the weights at each side of the balance. The amount of current needed was a measure of the weight on the beam. The balance was used as a load cell as shown in Fig. 1. On one side of the instrument a device was attached, which could support a hair sample at both ends and translate it vertically as well. The hook that in a usual TGS measurement holds the sample dish was now used to load the sample at the centre. On the other side of the lever a weight was attached. By lowering the supported ends (in which the hair was free to move) over a known distance, the sample was bent. The recorder of the system showed the difference in weight, which could be converted into force. The modulus of elasticity in bending Eb was derived from the acquired data using the relation: Eb =

l3 ∆F , 48I ∆δ

(1)

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433

Fig. 1. Schematic representation of the set-up for the three-point bending test.

where: ∆F = load increment as measured from preload, ∆δ = deflection increment at midspan as measured from preload, l = span length. For this experiment, for an elliptical sample the appropriate second moment of area I is given by: I=

π rmin 3 rmax , 4

(2)

where: rmin = minimum radius of the hair-sample, rmax = maximum radius of the hair-sample. The span length l was chosen to be 10 mm.

3. Material characterization 3.1. Experimental results Experiments have been performed on two different hairs at different values of regain. Each experiment was carried out three times to check the reproducibility. The minimum- and maximum radius of two samples of Asian hair were measured optically and are listed in Table 1. To reach the required regain, the whole experimental set-up was placed in a conditioned room. The temperature was kept constant at 20◦ C. Experiments were performed at a relative humidity of 2, 60, 80 and 99%. Since the water in hair affects the relaxation, the relative humidity was converted to the weight regain by using the sorption-model by D’Arcy and Watt [7], which was fitted to the desorption data of Chamberlain and Speakman [3]. This resulted in regain values of 1, 16, 21 and 31% respectively. A sample was put into water for one hour, after which it was hung straight in the above mentioned humidity conditions for two hours. Then the experiments were performed. The results are depicted in Fig. 2. As can be seen in this figure, the data clearly show a decrease of E(t) in time. Also, the initial values appear to vary with the regain.

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P. Zuidema et al. / The influence of humidity on the viscoelastic behaviour of human hair Table 1 Minimum and maximum radii of samples used rmin (µm) 47 58

Sample nr. 1 2

rmax (µm) 52 60

Fig. 2. Measurements of E as a function of time at different regain and their fits (- -), as described in Section 3.2. ∗ = 1%, + = 16%, ◦ = 21%, × = 31%.

3.2. Viscoelastic characterisation To describe the viscoelastic behaviour of human hair fibres a constitutive model has to be specified. It was decided to use the stretched exponent model to describe E(t) [12]: E(t) = (E0 − E∞ )Ψ(t) + E∞ ,

Ψ(t) = e−(t/τR ) , m

(3)

where: E0 = E∞ = τR = m=

initial modulus at t = 0, equilibrium modulus at t = ∞, relaxation time, variable which influences the relaxation rate.

To extend this model to other values of regain it was modified analogous to the reduced time principle [9]. This way, the stress resulting from a given strain is given by:
t

σ(t) = 0



 •



E Φ − Φ ε Φ dt .

(4)

Φ is defined as the reduced time: Φ(t) =


t 0

aRg0 (Rg ) dt

Φ (t ) =


t
0

aRg0 (Rg ) dt ,

(5)

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Table 2 Mean values of the derived parameters of the fitted curves in Fig. 2 Hair #1

Rg (%) 16 21 31

E0 –E∞ (GPa) 0.56 0.40 0.18

E∞ (GPa) 0.43 0.44 0.44

τ (s) 5200 3000 1600

Hair #2

1 0.68 0.39 139000 16 0.38 0.39 5100 21 0.28 0.39 3700 E0 = initial modulus, E∞ = equilibrium modulus, τ = time constant.

Fig. 3. Schematic illustration of the influence of regain.

where aRg0 (Rg ) is the shift-factor that describes the shift of time-scale as a function of regain (Rg ). It is both expected and observed that the regain will also have an influence on the E-modulus itself. This influence is described by a shift-factor bRg0 (Rg ). A least squares fitting procedure was used to fit the constitutive model to the data of Fig. 2 and the courses of the curves at different humidities were described. First results showed that m did not appear to vary significantly. Therefore it was chosen constant with a value of 0.4, and the fitting was repeated. The dotted lines in Fig. 2 represent these predictions. The parameters of the mean curves of Fig. 2 are presented in Table 2. The given values are the average of a few measurements (typically 2–3). The individual deviations from the average are in the order of 5–10% for the parameters E0 –E∞ and E∞ and in the order of 15–35% for τR . These predictions suggest that the equilibrium moduli for the different humidities are approximately the same. It is therefore assumed that, for all humidities, the equilibrium modulus remains the same for the same hair sample. The equilibrium modulus for different hair samples, however, can vary. This implies that the vertical shift-factor bRg0 (Rg ) is not proportional with E, but with E–E∞ , as is illustrated in Fig. 3. This changes Eq. (3) into: E(t) = bRg0 (Rg )(E0 − E∞ )Ψ(t) + E∞ ,

Ψ(t) = e−(Φ(t)/τR ) . m

(6)

The above conclusion is supported by the previously mentioned molecular relaxation mechanism of human hair fibres. In the extreme case that all hydrogen bonds are broken the E-modulus would reach its equilibrium modulus almost immediately, because its relaxation is not delayed by the hydrogen bonds. In the other extreme case, when all hydrogen bonds remain intact, the E-modulus will remain at its initial level, because no relaxation will occur. In all other cases, the equilibrium modulus will be reached after

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P. Zuidema et al. / The influence of humidity on the viscoelastic behaviour of human hair

Fig. 4. Shift-factors of both hairs in relation to the regain.

some period of time. Therefore, the humidity only influences the rate of relaxation and the magnitude of the initial modulus. 3.3. Application of time–humidity superposition The time–humidity superposition principle was applied to the measurements by shifting the curves at different humidities to the average curve at the state of Rg = 1%. Both a horizontal displacement log(aRg0 (Rg )), as well as a vertical proportionality bRg0 (Rg ) were used. From these shift-factors the reference curve at Rg = 0% was derived. The resulting shift-factors are presented in Fig. 4. To a first approximation a linear approach (forced through the origin) is chosen for easy modelling. Also the vertical proportionality was related to the reference state. The vertical proportionality-factors appeared not to be the same for both hairs. This is probably caused by the value of the equilibrium moduli of both hairs in relation to their initial value, which differ significantly from each other. In order to compare both hairs, their equilibrium moduli E∞ were subtracted from their measured values of E(t) and subsequently normalised round the resulting initial value (E0 –E∞ ) at the reference state. This resulted in a dimensionless variable K1 , ranging from 1 (at Rg = 0%) to 0: K1 =

E(t) − E∞ . E0 − E∞

(7)

This way, for all data of both hairs at the same conditions new proportionality-factors were derived. These factors showed a good linearity with the amount of regain, as depicted in Fig. 5. It is therefore assumed that K1 depends linearly on the regain and has the value of 1 at a regain of 0. 4. Validation of the model 4.1. Experimental To verify the predictions of the model experiments were carried out during which a single hair fibre was curled by winding it around a mandrel with a radius of 2.5 mm. A groove determined the thread of the

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437

Fig. 5. Proportionality-factors in relation to the humidity.

curl, which was set to 0.25 mm. The mandrel with the fibre around it was placed in water for 5 minutes, after which it was conditioned for one hour in air, with a relative humidity of 60%. The temperature was kept constant at T = 20◦ C. After conditioning the hair fibre was hung vertically. The radius of the helix and its thread were measured during one hour. From this the curvature K of the fibre was derived by: K=

R2

R , + S 2 /4π 2

(8)

where R represents the radius of the curl and S represents the thread of the curl. The level of set was determined by the quotient of the curvature at time t and the initial curvature (which is the curvature of the mandrel): set =

K(t) . Kmandrel

(9)

The experiment was carried out for six different hairs. The results are discussed in Section 4.3, along with the theoretical prediction, which will be described next. 4.2. Predicting the level of set using the derived model Using the derived formulation, it is possible to predict the level of set of a curled hair after release, as well as the course of the level of set. The level of set is determined by the remaining level of set at time t: set =

ε(t) . ε0

(10)

The simulated conditioning was carried out in the same way as the experiment, as illustrated in Fig. 6A. After one hour, the regain has virtually reached its final amount of regain. Therefore, the regain is assumed to be constant after one hour. Also after one hour, the forced deformation was removed, which, because of the stress relaxation, results in a remaining level of set. This level of set will decrease in time as a result of recovery within the hair fibre. This is illustrated in Fig. 6B.

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P. Zuidema et al. / The influence of humidity on the viscoelastic behaviour of human hair

Fig. 6. Illustration of the courses of regain (A) and set (B) during conditioning, set and release.

To determine the reduced time Φ [9], the regain of the human hair fibre as a function of time has to be known. For this purpose a theoretical model of the drying of hair by Ackermans [1] was used. This model gives a description of the course of the regain in time, starting from saturated hair. The principles of the model were described for wool by Augustin [2]. It has been extended to fit the case of drying hair. The basic idea is that each of the three types of water in the hair (as appear in the D’Arcy–Watt sorption equation [7]) has its own rate to change from one concentration to the other. Combining the three gives the drying curve for hair. The reduced time is determined by dividing the first hour of the experiment into short time-steps. At each time-step the mean value of the regain is calculated, and the contribution to the reduced time is determined. The resulting total reduced time Φ is used as the setting time for a hair in the reference state (T = 20◦ C; Rg = 0%). Using this setting time, the relaxation after one hour is calculated. When the hair is released, the amount of set is determined after each minute. 4.3. Results Both the predicted values and the measured values of the level of set are given in Fig. 7. In this figure, at t = 0, the forced deformation is removed and the initial level of set is instantly reached. After t = 0, the measured level of set proves to be approximately the same as the predicted level of set. The rate of set-loss also shows reasonable agreement, although the predicted rate appears to be slightly higher. This slight difference could be related to the influence of progressive physical aging. The hair is initially wet, leading to a value of the a glass transition temperature (Tg ) which is below room temperature. During the drying process the Tg of the hair is raised above room temperature [14] and physical aging is likely to be initiated. As physical aging delays the viscoelastic response of the material, it could rationalise the lower rate of set-loss observed in the experiments. Nevertheless, the results indicate that the model proposed contains the necessary elements to predict the initial set of hair and its course in time from the setting process. 5. Conclusions The superposition principle seems to be applicable for time–humidity equivalence. The derived relationships are in accordance with the molecular theory, which considerably enhances their credibility.

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Fig. 7. Predicted set (-) and measurements versus time.

However, more tests on different hair samples should be done to confirm this theory, as well as to derive these relationships with better accuracy. The general characterisation that was derived can be used to describe the viscoelastic behaviour of human hair fibres. From the comparison between measured set and theoretical set it can be concluded that the derived characterisation provides a tool to predict both the level of set and recovery during curling. References [1] P.A.J. Ackermans, 11th International Hair-Science Symposium, Drying behaviour of human hair, Maastricht, 9–11 September 1998. [2] P. Augustin, Adsorption und desorption von Wasser in und aus Keratinfasern, Ph.D. Dissertation, Rheinisch-Westfälischen Technische Hochschule Aachen, 1996. [3] N.H. Chamberlain and J.B. Speakman, Über Hysteresiserscheinungen in der Wasserafname des Menschenhaares, Z. Electrochem. 37 (1931), 374–377. [4] B.M. Chapman, Linear superposition of viscoelastic responses in nonequilibrium systems, J. Appl. Polym. Sci. 18 (1974), 3523–3536. [5] B.M. Chapman, Linear superposition of time-variant viscoelastic responses, J. Phys. D 7 (1974), L185–188. [6] B.M. Chapman, The bending and recovery of fabrics under conditions of changing temperature and relative humidity, Text. Res. J. 46 (1976), 113–122. [7] R.L. D’Arcy and I.C. Watt, Analysis of sorption isotherms of non-homogeneous sorbents, Trans. Faraday Soc. 66 (1970), 1236–1245. [8] M. Feughelman, Mechanical Properties and Structure of Alpha-Keratin Fibres, UNSW Press, Sydney, 1997. [9] J.D. Ferry, Viscoelastic Properties of Polymers, 3rd edn, John Wiley & Sons, New York, 1980. [10] W.E. Morton and J.W.S. Hearle, Physical Properties of Textile Fibres, 3rd edn, The Textile Institute, Manchester, 1993. [11] L.C.E. Struik, Physical Aging in Amorphous Polymers and Other Materials, Elsevier, Amsterdam, 1978. [12] G. Williams and D.C. Watt, Non-symmetrical dielectric Behaviour arising from a simple empirical decay function, Trans. Faraday Soc. 66 (1970), 80–85. [13] F.-J. Wortmann, The influence of water on the viscoelastic properties of wool fibres, Textile Res. J. 55 (1985), 750–756. [14] F.-J. Wortmann, B.J. Rigby and D.G. Philips, Glass transition temperature of wool as a function of regain, Textile Res. J. 54 (1984), 6–8.